aa r X i v : . [ m a t h . A T ] J un More stable stems
Daniel C. IsaksenGuozhen WangZhouli Xu
Author address:
Department of Mathematics, Wayne State University, Detroit, MI48202, USA
E-mail address : [email protected] Shanghai Center for Mathematical Sciences, Fudan University,Shanghai, China, 200433
E-mail address : [email protected] Department of Mathematics, Massachusetts Institute of Technol-ogy, Cambridge, MA 02139
E-mail address : [email protected] ontents List of Tables ixChapter 1. Introduction 11.1. New Ingredients 21.2. Main results 41.3. Remaining uncertainties 91.4. Groups of homotopy spheres 91.5. Notation 101.6. How to use this manuscript 121.7. Acknowledgements 12Chapter 2. Background 132.1. Associated graded objects 132.2. Motivic modular forms 162.3. The cohomology of the C -motivic Steenrod algebra 172.4. Toda brackets 17Chapter 3. The algebraic Novikov spectral sequence 253.1. Naming conventions 253.2. Machine computations 263.3. h -Bockstein spectral sequence 27Chapter 4. Massey products 294.1. The operator g d differential 335.2. The Adams d differential 345.3. The Adams d differential 405.4. The Adams d differential 455.5. Higher differentials 48Chapter 6. Toda brackets 53Chapter 7. Hidden extensions 597.1. Hidden τ extensions 597.2. Hidden 2 extensions 617.3. Hidden η extensions 697.4. Hidden ν extensions 76 vi CONTENTS bstract We compute the stable homotopy groups up to dimension 90, except for somecarefully enumerated uncertainties.
Mathematics Subject Classification.
Primary 14F42, 55Q45, 55S10, 55T15; Secondary16T05, 55P42, 55Q10, 55S30 .
Key words and phrases. stable homotopy group, stable motivic homotopy theory, May spec-tral sequence, Adams spectral sequence, cohomology of the Steenrod algebra, Adams-Novikovspectral sequence.The first author was supported by NSF grant DMS-1606290. The second author was sup-ported by grant NSFC-11801082. The third author was supported by NSF grant DMS-1810638.Many of the associated machine computations were performed on the Wayne State UniversityGrid high performance computing cluster. vii ist of Tables π ∗ , ∗
912 Some hidden values of the unit map of mmf
923 Some Massey products in Ext C E -page generators of the C -motivic Adams spectral sequence 945 Some permanent cycles in the C -motivic Adams spectral sequence 1036 Adams d differentials in the C -motivic Adams spectral sequence 1047 Adams d differentials in the C -motivic Adams spectral sequence 1078 Adams d differentials in the C -motivic Adams spectral sequence 1099 Higher Adams differentials in the C -motivic Adams spectral sequence 10910 Unknown Adams differentials in the C -motivic Adams spectral sequence 11011 Some Toda brackets 11112 Hidden values of inclusion of the bottom cell into Cτ Cτ to the top cell 11414 Hidden τ extensions in the C -motivic Adams spectral sequence 11615 Hidden 2 extensions 11816 Possible hidden 2 extensions 11917 Hidden η extensions 12018 Possible hidden η extensions 12219 Hidden ν extensions 12320 Possible hidden ν extensions 12421 Miscellaneous hidden extensions 125 ix HAPTER 1
Introduction
The computation of stable homotopy groups of spheres is one of the most fun-damental and important problems in homotopy theory. It has connections to manytopics in topology, such as the cobordism theory of framed manifolds, the classifica-tion of smooth structures on spheres, obstruction theory, the theory of topologicalmodular forms, algebraic K-theory, motivic homotopy theory, and equivariant ho-motopy theory.Despite their simple definition, which was available eighty years ago, thesegroups are notoriously hard to compute. All known methods only give a completeanswer through a range, and then reach an obstacle until a new method is intro-duced. The standard approach to computing stable stems is to use Adams typespectral sequences that converge from algebra to homotopy. In turn, to identifythe algebraic E -pages, one needs algebraic spectral sequences that converge fromsimpler algebra to more complicated algebra. For any spectral sequence, difficul-ties arise in computing differentials and in solving extension problems. Differentmethods lead to trade-offs. One method may compute some types of differentialsand extension problems efficiently, but leave other types unanswered, perhaps evenunsolvable by that technique. To obtain complete computations, one must be eclec-tic, applying and combining different methodologies. Even so, combining all knownmethods, there are eventually some problems that cannot be solved. Mahowald’suncertainty principle states that no finite collection of methods can completelycompute the stable homotopy groups of spheres.Because stable stems are finite groups (except for the 0-stem), the computationis most easily accomplished by working one prime at a time. At odd primes, theAdams-Novikov spectral sequence and the chromatic spectral sequence, which arebased on complex cobordism and formal groups, have yielded a wealth of data [ ].As the prime grows, so does the range of computation. For example, at the primes3 and 5, we have complete knowledge up to around 100 and 1000 stems respectively[ ].The prime 2, being the smallest prime, remains the most difficult part of thecomputation. In this case, the Adams spectral sequence is the most effective tool.The manuscript [ ] presents a careful analysis of the Adams spectral sequence, inboth the classical and C -motivic contexts, that is essentially complete through the59-stem. This includes a verification of the details in the classical literature [ ] [ ][ ] [ ]. Subsequently, the second and third authors computed the 60-stem and61-stem [ ].We also mention [ ] [ ], which take an entirely different approach to com-puting stable homotopy groups. However, the computations in [ ] [ ] are nowknown to contain several errors. See [ , Section 2] for a more detailed discussion. The goal of this manuscript is to continue the analysis of the Adams spectralsequence into higher stems at the prime 2. We will present information up to the90-stem. While we have not been able to resolve all of the possible differentialsin this range, we enumerate the handful of uncertainties explicitly. See especiallyTable 10 for a summary of the possible differentials that remain unresolved.The charts in [ ] and [ ] are an essential companion to this manuscript.They present the same information in an easily interpretable graphical format.Our analysis uses various methods and techniques, including machine-generatedhomological algebra computations, a deformation of homotopy theories that con-nects C -motivic and classical stable homotopy theory, and the theory of motivicmodular forms. Here is a quick summary of our approach:(1) Compute the cohomology of the C -motivic Steenrod algebra by machine.These groups serve as the input to the C -motivic Adams spectral sequence.(2) Compute by machine the algebraic Novikov spectral sequence that con-verges to the cohomology of the Hopf algebroid ( BP ∗ , BP ∗ BP ). Thisincludes all differentials, and the multiplicative structure of the cohomol-ogy of ( BP ∗ , BP ∗ BP ).(3) Identify the C -motivic Adams spectral sequence for the cofiber of τ withthe algebraic Novikov spectral sequence [ ]. This includes an identifica-tion of the cohomology of ( BP ∗ , BP ∗ BP ) with the homotopy groups ofthe cofiber of τ .(4) Pull back and push forward Adams differentials for the cofiber of τ toAdams differentials for the C -motivic sphere, along the inclusion of thebottom cell and the projection to the top cell.(5) Deduce additional Adams differentials for the C -motivic sphere with avariety of ad hoc arguments. The most important methods are Todabracket shuffles and comparison to the motivic modular forms spectrum mmf [ ].(6) Deduce hidden τ extensions in the C -motivic Adams spectral sequence forthe sphere, using a long exact sequence in homotopy groups.(7) Obtain the classical Adams spectral sequence and the classical stable ho-motopy groups by inverting τ .The machine-generated data that we obtain in steps (1) and (2) are availableat [ ]. See also [ ] for a discussion of the implementation of the machine com-putation.Much of this process is essentially automatic. The exception occurs in step (5)where ad hoc arguments come into play.This document describes the results of this systematic program through the 90-stem. We anticipate that our approach will allow us to compute into even higherstems, especially towards the last unsolved Kervaire invariant problem in dimension126. However, we have not yet carried out a careful analysis. We discuss in more detail several new ingredients that allow us to carry outthis program.
The Adams-Novikov spectralsequence has been used very successfully to carry out computations at odd primes. .1. NEW INGREDIENTS 3
However, at the prime 2, its usage has not been fully exploited in stemwise com-putations. This is due to the difficulty of computing its E -page. The first authorpredicted in [ ] that “the next major breakthrough in computing stable stems willinvolve machine computation of the Adams-Novikov E -page.”The second author achieved this machine computation; the resulting data isavailable at [ ]. The process goes roughly like this. Start with a minimal reso-lution that computes the cohomology of the Steenrod algebra. Lift this resolutionto a resolution of BP ∗ BP . Finally, use the Curtis algorithm to compute the ho-mology of the resulting complex, and to compute differentials in the associatedalgebraic spectral sequences, such as the algebraic Novikov spectral sequence andthe Bockstein spectral sequence. See [ ] for further details. The C -motivic stable homotopy cate-gory gives rise to new methods to compute stable stems. These ideas are used in acritical way in [ ] to compute stable stems up to the 59-stem.The key insight of this article that distinguishes it significantly from [ ] isthat C -motivic cellular stable homotopy theory is a deformation of classical stablehomotopy theory [ ]. From this perspective, the “generic fiber” of C -motivicstable homotopy theory is classical stable homotopy theory, and the “special fiber”has an entirely algebraic description. The special fiber is the category of BP ∗ BP -comodules, or equivalently, the category of quasicoherent sheaves on the modulistack of 1-dimensional formal groups.In more concrete terms, let Cτ be the cofiber of the C -motivic stable map τ .The homotopy category of Cτ -modules has an algebraic structure [ ]. In partic-ular, the C -motivic Adams spectral sequence for Cτ is isomorphic to the algebraicNovikov spectral sequence that computes the E -page of the Adams-Novikov spec-tral sequence. Using naturality of Adams spectral sequences, the differentials inthe algebraic Novikov spectral sequence, which are computed by machine, can belifted to differentials in the C -motivic Adams spectral sequence for the C -motivicsphere spectrum. Then the Betti realization functor produces differentials in theclassical Adams spectral sequence.Our use of C -motivic stable homotopy theory appears to rely on the funda-mental computations, due to Voevodsky [ ] [ ], of the motivic cohomology of apoint and of the motivic Steenrod algebra. In fact, recent progress has determinedthat our results do not depend on this deep and difficult work. There are nowpurely topological constructions of homotopy categories that have identical com-putational properties to the cellular stable C -motivic homotopy category [ ] [ ].In these homotopy categories, one can obtain from first principles the fundamentalcomputations of the cohomology of a point and of the Steenrod algebra, using onlywell-known classical computations. Therefore, the material in this manuscript doesnot logically depend on Voevodsky’s work, even though the methods were verymuch inspired by his groundbreaking computations. In classical chromatic homotopy theory, thetheory of topological modular forms, introduced by Hopkins and Mahowald [ ],plays a central role in the computations of the K (2)-local sphere.Using a topological model of the cellular stable C -motivic homotopy category,one can construct a “motivic modular forms” spectrum mmf [ ], whose motivic
1. INTRODUCTION cohomology is the quotient of the C -motivic Steenrod algebra by its subalgebra gen-erated by Sq , Sq , and Sq . Just as tmf plays an essential role in studies of the clas-sical Adams spectral sequence, mmf is an essential tool for motivic computations.The C -motivic Adams spectral sequence for mmf can be analyzed completely [ ],and naturality of Adams spectral sequences along the unit map of mmf providesmuch information about the behavior of the C -motivic Adams spectral sequencefor the C -motivic sphere spectrum. We summarize our main results in the following theorem and corollaries.
Theorem . The C -motivic Adams spectral sequence for the C -motivic spherespectrum is displayed in the charts in [ ] , up to the -stem. The proof of Theorem 1.1 consists of a series of specific computational facts,which are verified throughout this manuscript.
Corollary . The classical Adams spectral sequence for the sphere spectrumis displayed in the charts in [ ] , up to the -stem. Corollary 1.2 follows immediately from Theorem 1.1. One simply inverts τ , orequivalently ignores τ -torsion. Corollary . The Adams-Novikov spectral sequence for the sphere spectrumis displayed in the charts in [ ] . Corollary 1.3 also follows immediately from Theorem 1.1. As described in[ , Chapter 6], the Adams-Novikov spectral sequence can be reverse-engineeredfrom information about C -motivic stable homotopy groups. Corollary . Table 1 describes the stable homotopy groups π k for all valuesof k up to . We adopt the following notation in Table 1. An integer n stands for the cyclicabelian group Z /n ; the symbol · by itself stands for the trivial group; the expression n · m stands for the direct sum Z /n ⊕ Z /m ; and n j stands for the direct sum of j copies of Z /n . The horizontal line after dimension 61 indicates the range in whichour computations are new information.Table 1 describes each group π k as the direct sum of three subgroups: the2-primary v -torsion, the odd primary v -torsion, and the v -periodic subgroups.The last column of Table 1 describes the groups of homotopy spheres thatclassify smooth structures on spheres in dimensions at least 5. See Section 1.4 andTheorem 1.7 for more details.Starting in dimension 82, there remain some uncertainties in the 2-primary v -torsion. In most cases, these uncertainties mean that the order of some stablehomotopy groups are known only up to factors of 2. In a few cases, the additivegroup structures are also undetermined.These uncertainties have two causes. First, there are a handful of differentialsthat remain unresolved. Table 10 describes these. Second, there are some possiblehidden 2 extensions that remain unresolved. .2. MAIN RESULTS 5 Table 1: Stable homotopy groups up to dimension 90 k v -torsion v -torsion v -periodic group ofat the prime 2 at odd smooth structuresprimes1 · · · · · · · · · · · · · ?5 · · · · · · · · · · · b · · · · · · · · · b · · · · · ·
314 2 · · ·
215 2 · · · b ·
216 2 · · ·
18 8 · ·
819 2 · · · b ·
220 8 3 · ·
321 2 · · ·
22 2 · ·
23 2 · · · · · b · · ·
324 2 · · · ·
226 2 3 2 2 · · · · b
28 2 · · · ·
330 2 3 ·
331 2 · · · · b ·
32 2 ·
33 2 · ·
34 2 · · ·
435 2 · · · · b ·
36 2 3 · ·
337 2 · · ·
338 2 · · · · · ·
539 2 · · · b · ·
340 2 · · ·
341 2 · ·
42 2 · · ·
1. INTRODUCTION
Table 1: Stable homotopy groups up to dimension 90 k v -torsion v -torsion v -periodic group ofat the prime 2 at odd smooth structuresprimes43 · · · · b
44 8 · ·
845 2 ·
16 9 · · · · · ·
546 2 · ·
347 2 · · · · · b · · ·
348 2 · · · · · ·
350 2 ·
351 2 · · · b · ·
852 2 · ·
353 2 · · ·
54 2 · · · · · · · · b · · · ·
57 2 · ·
58 2 ·
59 2 · · · · · b ·
60 4 · · · · · ·
62 2 · ·
363 2 · · · · · b · ·
464 2 · · ·
465 2 · · · ·
366 2 · · ·
867 2 · · · b · ·
468 2 · ·
369 2 · · ·
70 2 · · · ·
71 2 · · · · · · · · · b · · ·
872 2 ·
373 2 · ·
74 4 · ·
375 2 9 8 · b · ·
976 2 · · · ·
577 2 · · · · ·
478 2 · · · ·
379 2 · · · · · · b · ·
480 2 ·
81 2 · · · · · ·
82 2 · · · · · · · · · · · · · · ·
783 2 · · · · · b · · · b · · ·
584 2 or 2 · · or 2 · Table 1: Stable homotopy groups up to dimension 90 k v -torsion v -torsion v -periodic group ofat the prime 2 at odd smooth structuresprimes85 2 · or 2 · or 3 · · · or 2 · · · or 2 · ·
86 2 · or 2 · or 3 · · · · · · · · · · or 2 · · · · · · · · · ·
587 2 or 2 or · · · · b · or b · or2 · · b · · b · ·
488 2 · · ·
489 2 · ·
90 2 · · · · · · n dots indicate Z / n . Thenon-vertical lines indicate multiplications by η and ν . The blue dots represent the v -periodic subgroups. The green dots are associated to the topological modularforms spectrum tmf . These elements are detected by the unit map from the spherespectrum to tmf , either in homotopy or in the algebraic Ext groups that serve asAdams E -pages.Finally, the red dots indicate uncertainties. In addition, in higher stems, thereare possible extensions by 2, η , and ν that are not indicated in Figure 1. See Tables16, 18, and 20 for more details about these possible extensions. Figure 1.
1. INTRODUCTION
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30012345678
30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60
31 32 33 34 35 36 37 38 39 40 41 42 43 44 4501234567891011
62 64 66 68 70 72 74 76 78 80 82 84 86 88 90The orders of individual 2-primary stable homotopy groups do not follow aclear pattern, with large increases and decreases seemingly at random. However,an empirically observed pattern emerges if we consider the cumulative size of thegroups, i.e., the product of the orders of all 2-primary stable homotopy groups fromdimension 1 to dimension k . .4. GROUPS OF HOMOTOPY SPHERES 9 Our data strongly suggest that asymptotically, there is a linear relationship be-tween k and the logarithm of this product of orders. In other words, the numberof dots in Figure 1 in stems 1 through k is linearly proportional to k . Correspond-ingly, the number of dots in the classical Adams E ∞ -page in stems 1 through k is linearly proportional to k . Thus, in extending from dimension 60 to dimension90, the overall size of the computation more than doubles. Specifically, through di-mension 60, the cumulative rank of the Adams E ∞ -page is 199, and is 435 throughdimension 90. Similarly, through dimension 60, the cumulative rank of the Adams E -page is 488, and is 1,461 through dimension 90. Conjecture . Let f ( k ) be the product of the orders of the -primary stablehomotopy groups in dimensions through k . There exists a non-zero constant C such that lim k →∞ log f ( k ) k = C. One interpretation of this conjecture is that the expected value of the logarithmof the order of the 2-primary component of π k grows linearly in k . We have only datato support the conjecture, and we have not formulated a mathematical rationale.It is possible that in higher stems, new phenomena occur that alter the growth rateof the stable homotopy groups.By comparison, data indicates that the growth rate of the Adams E -page isqualitatively greater than the growth rate of the Adams E ∞ -page. This apparentmismatch has implications for the frequency of Adams differentials. Some uncertainties remain in the analysis of the first 90 stable stems. Table10 lists all possible differentials in this range that are undetermined. Some ofthese unknown differentials are inconsequential because possible error terms arekilled by later differentials. Others are inconsequential because they only affect thenames (and Adams filtrations) of the elements in the Adams E ∞ -page that detectparticular stable homotopy elements. A few of the unknown differentials affect thestructure of the stable homotopy groups more significantly. This means that theorders of some of the stable homotopy groups are known only up to factors of 2.In addition, there are some possible hidden extensions by 2, η , and ν that re-main unresolved. Tables 16, 18, and 20 summarize these possibilities. The presenceof unknown hidden extensions by 2 means that the group structures of some stablehomotopy groups are not known, even though their orders are known. An important application of stable homotopy group computations is to thework of Kervaire and Milnor [ ] on the classification of smooth structures onspheres in dimensions at least 5. Let Θ n be the group of h -cobordism classes ofhomotopy n -spheres. This group classifies the differential structures on S n for n ≥
5. It has a subgroup Θ bpn , which consists of homotopy spheres that boundparallelizable manifolds. The relation between Θ n and the stable homotopy group π n is summarized in Theorem 1.6. See also [ ] for a survey on this subject. Theorem . (Kervaire-Milnor [ ] ) Suppose that n ≥ . (1) The subgroup Θ bpn is cyclic, and has the following order: | Θ bpn | = , if n is even, or , if n = 4 k + 1 ,b k , if n = 4 k − . Here b k is k − (2 k − − times the numerator of B k /k , where B k isthe k th Bernoulli number.(2) For n mod , there is an exact sequence / / Θ bpn / / Θ n / / π n /J / / . Here π n /J is the cokernel of the J -homomorphism.(3) For n ≡ mod , there is an exact sequence / / Θ bpn / / Θ n / / π n /J Φ / / Z / / / Θ bpn − / / . Here the map Φ is the Kervaire invariant. The first few values, and then estimates, of the numbers b k are given by thesequence28 , , , , . × , . × , . × , . × , . . . . Theorem . The last column of Table 1 describes the groups Θ n for n ≤ ,with the exception of n = 4 . The underlined symbols denote the contributions from Θ bpn . The cokernel of the J -homomorphism is slightly different than the v -torsionpart of π n at the prime 2. In dimensions 8 m + 1 and 8 m + 2, there are classesdetected by P m h and P m h in the Adams spectral sequence. These classes are v -periodic, in the sense that they are detected by the K (1)-local sphere. However,they are also in the cokernel of the J -homomorphism.We restate the following conjecture from [ ], which is based on the currentknowledge of stable stems and a problem proposed by Milnor [ ]. Conjecture . In dimensions greater than 4, the only spheres with uniquesmooth structures are S , S , S , S , and S . Uniqueness in dimensions 5, 6 and 12 was known to Kervaire and Milnor [ ].Uniqueness in dimension 56 is due to the first author [ ], and uniqueness in di-mension 61 is due to the second and the third authors [ ].Conjecture 1.8 is equivalent to the claim that the group Θ n is not of order1 for dimensions greater than 61. This conjecture has been confirmed in all odddimensions by the second and the third authors [ ] based on the work of Hill,Hopkins, and Ravenel [ ], and in even dimensions up to 140 by Behrens, Hill,Hopkins, and Mahowald [ ]. The cohomology of the Steenrod algebra is highly irregular, so consistent nam-ing systems for elements presents a challenge. A list of multiplicative generators .5. NOTATION 11 appears in Table 4. To a large extent, we rely on the traditional names for ele-ments, as used in [ ], [ ], [ ], and elsewhere. However, we have adopted somenew conventions in order to partially systematize the names of elements.First, we use the symbol ∆ x to indicate an element that is represented by v x in the May spectral sequence. This use of ∆ is consistent with the role that v plays in the homotopy of tmf , where it detects the discriminant element ∆. Forexample, instead of the traditional symbol r , we use the name ∆ h .Second, the symbol M indicates the Massey product operator h− , h , g i . Forexample, instead of the traditional symbol B , we use the name M h .Similarly, the symbol g indicates the Massey product operator h− , h , h i . Forexample, we write h g for the indecomposable element h h , h , h i .Eventually, we encounter elements that neither have traditional names, nor canbe named using symbols such as P , ∆, M , and g . In these cases, we use arbitrarynames of the form x s,f , where s and f are the stem and Adams filtration of theelement.The last column of Table 4 gives alternative names, if any, for each multiplica-tive generator. These alternative names appear in at least one of [ ] [ ] [ ]. Remark . One specific element deserves further discussion. We define τ Q to be the unique element such that h · τ Q = 0. This choice is not compatible withthe notation of [ ]. The element τ Q from [ ] equals the element τ Q + τ n inthis manuscript.We shall also extensively study the Adams spectral sequence for the cofiber of τ . See Section 3.1 for more discussion of the names of elements in this spectralsequence, and how they relate to the Adams spectral sequence for the sphere.Table 1 gives some notation for elements in π ∗ , ∗ . Many of these names followstandard usage, but we have introduced additional non-standard elements such as κ and κ . These elements are defined by the classes in the Adams E ∞ -page thatdetect them. In some cases, this style of definition leaves indeterminacy becauseof the presence of elements in the E ∞ -page in higher filtration. In some of thesecases, Table 1 provides additional defining information. Beware that this additionaldefining information does not completely specify a unique element in π ∗ , ∗ in allcases. For the purposes of our computations, these remaining indeterminacies arenot consequential.(1) Cτ represents the cofiber of τ : S , − → S , . We can also write S/τ forthis C -motivic spectrum, but the latter notation is more cumbersome.(2) Ext = Ext C is the cohomology of the C -motivic Steenrod algebra. It isgraded in the form ( s, f, w ), where s is the stem (i.e., the total degree mi-nus the Adams filtration), f is the Adams filtration (i.e., the homologicaldegree), and w is the motivic weight.(3) Ext cl is the cohomology of the classical Steenrod algebra. It is graded inthe form ( s, f ), where s is the stem (i.e., the total degree minus the Adamsfiltration), and f is the Adams filtration (i.e., the homological degree).(4) π ∗ , ∗ is the 2-completed C -motivic stable homotopy groups.(5) H ∗ ( S ; BP ) is the Adams-Novikov E -page for the classical sphere spec-trum, i.e., Ext BP ∗ BP ( BP ∗ , BP ∗ ).(6) H ∗ ( S/ BP ) is the Adams-Novikov E -page for the classical mod 2 Moorespectrum, i.e., Ext BP ∗ BP ( BP ∗ , BP ∗ / The manuscript is oriented around a series of tables to be found in Chapter 8.In a sense, the rest of the manuscript consists of detailed arguments for establishingeach of the computations listed in the tables. We have attempted to give referencesand cross-references within these tables, so that the reader can more easily find thespecific arguments pertaining to each computation.We have attempted to make the arguments accessible to users who do not intendto read the manuscript in its entirety. To some extent, with an understanding ofhow the manuscript is structured, it is possible to extract information about aparticular homotopy class in isolation.We assume that the reader is also referring to the Adams charts in [ ] and[ ]. These charts describe the same information as the tables, except in graphicalform.This manuscript is very much a sequel to [ ]. We will frequently refer todiscussions in [ ], rather than repeat that same material here in an essentiallyredundant way. This is especially true for the first parts of Chapters 2, 3, and 4of [ ], which discuss respectively the general properties of Ext, the May spectralsequence, and Massey products; the Adams spectral sequence and Toda brackets;and hidden extensions.Chapter 2 provides some additional miscellaneous background material not al-ready covered in [ ]. Chapter 3 discusses the nature of the machine-generateddata that we rely on. In particular, it describes our data on the algebraic Novikovspectral sequence, which is equal to the Adams spectral sequence for the cofiber of τ . Chapter 4 provides some tools for computing Massey products in Ext, and givessome specific computations. Chapter 5 carries out a detailed analysis of Adams dif-ferentials. Chapter 6 computes some miscellaneous Toda brackets that are neededfor various specific arguments elsewhere. Chapter 7 methodically studies hiddenextensions by τ , 2, η , and ν in the E ∞ -page of the C -motivic Adams spectral se-quence. This chapter also gives some information about other miscellaneous hiddenextensions. Finally, Chapter 8 includes the tables that summarize the multitude ofspecific computations that contribute to our study of stable homotopy groups. We thank Agnes Beaudry, Mark Behrens, Robert Bruner, Robert Burklund,Dexter Chua, Paul Goerss, Jesper Grodal, Lars Hesselholt, Mike Hopkins, PeterMay, Haynes Miller, Christian Nassau, Doug Ravenel, and John Rognes for theirsupport and encouragement throughout this project.HAPTER 2
Background
Definition . A filtered object A consists of a finite chain A = F A ⊇ F A ⊇ F A ⊇ · · · ⊇ F p − A ⊇ F p A = 0of inclusions descending from A to 0.We will only consider finite chains because these are the examples that arisein our Adams spectral sequences. Thus we do not need to refer to “exhaustive”and “Hausdorff” conditions on filtrations, and we avoid subtle convergence issuesassociated with infinite filtrations. Example . The C -motivic stable homotopy group π , = Z / ⊕ Z / σ and κ . The subgroup F is zero, the subgroup F = F is generated by κ , and thesubgroup F = F = F is generated by σ and κ . Definition . Let A be a filtered object. The associated graded object Gr A is p M F i A/F i +1 A. If a is an element of Gr A , then we write { a } for the set of elements of A that are detected by a . In general, { a } consists of more than one element of A ,unless a happens to have maximal filtration. More precisely, the element a is acoset α + F i +1 A for some α in A , and { a } is another name for this coset. In thissituation, we say that a detects α .In this manuscript, the main example of a filtered object is a C -motivic homo-topy group π p,q , equipped with its Adams filtration. Example . Consider the C -motivic stable homotopy group π , with itsAdams filtration, as described in Example 2.2. The associated graded object isnon-trivial only in degrees 2 and 4, and it is generated by h and d respectively. Definition . Let A and B be filtered objects, perhaps with filtrations ofdifferent lengths. A map f : A → B is filtration preserving if f ( F i A ) is containedin F i B for all i .Let f : A → B be a filtration preserving map of filtered objects. We writeGr f : Gr A → Gr B for the induced map on associated graded objects. Definition . Let a and b be elements of Gr A and Gr B respectively. Wesay that b is the (not hidden) value of a under f if Gr f ( a ) = b .We say that b is the hidden value of a under f if:
134 2. BACKGROUND (1) Gr f ( a ) = 0.(2) there exists an element α of { a } in A such that f ( α ) is contained in { b } in B .(3) there is no element γ in filtration strictly higher than α such that f ( γ ) iscontained in { b } .The motivation for condition (3) may not be obvious. The point is to avoidsituations in which condition (2) is satisfied trivially. Suppose that there is anelement γ such that f ( γ ) is contained in { b } . Let a be any element of Gr A whosefiltration is strictly less than the filtration of γ . Now let α be any element of { a } such that f ( α ) = 0. (It may not be possible to choose such an α in general, butsometimes it is possible.) Then α + γ is another element of { a } such that f ( α + γ )is contained in { b } . Thus f takes some element of { a } into { b } , but only becauseof the presence of γ . Condition (3) is designed to exclude this situation. Example . We illustrate the role of condition (3) in Definition 2.6 with aspecific example. Consider the map η : π , → π , . The associated graded mapGr( η ) takes h to 0 and takes d to h d .The coset { h } in π , consists of two elements σ and σ + κ . One of theseelements is non-zero after multiplying by η . (In fact, ησ equals zero, and η ( σ + κ ) = ηκ is non-zero, but that is not relevant here.) Conditions (1) and (2) ofDefinition 2.6 are satisfied, but condition (3) fails because of the presence of κ inhigher filtration.Suppose that b is the hidden value of a under f . It is typically the case that f ( α ) is contained in { b } for every α in A . However, an even more complicatedsituation can occur in which this is not true.Suppose that b is the hidden value of a under f , and suppose that b is the(hidden or not hidden) value of a under f . Moreover, suppose that the filtrationof a is strictly lower than the filtration of a , and the filtration of b is strictlygreater than the filtration of b . In this situation, we say that the value of a under f crosses the value of a under f .The terminology arises from the usual graphical calculus, in which elements ofhigher filtration are drawn above elements of lower filtration, and values of mapsare indicated by line segments, as in Figure 1. Figure 1.
Crossing values a a b b Example . For any map X → Y of C -motivic spectra, naturality of theAdams spectral sequence induces a filtration preserving map π p,q X → π p,q Y . We .1. ASSOCIATED GRADED OBJECTS 15 are often interested in inclusion S , → Cτ of the bottom cell into Cτ , and inprojection Cτ → S , − from Cτ to the top cell. We also consider the unit map S , → mmf . Definition 2.6 allows for the pos-sibility of some essentially redundant cases. In order to avoid this redundancy, weintroduce indeterminacy into our definition.Suppose, as in Definition 2.6, that b is the hidden value of a under f , so thereexists some α in { a } such that f ( α ) is contained in { b } . Suppose also that thereis another element a ′ in Gr A in degree strictly greater than the degree of a , suchthat f ( α ′ ) is contained in { b ′ } , where α ′ is in { a ′ } and b ′ has the same degree as b . Then b + b ′ is also a hidden value of a under f , since α + α ′ is contained in { a } and f ( α + α ′ ) is contained in { b + b ′ } . In this case, we say that b ′ belongs to thetarget indeterminacy of the hidden value. Example . Consider the map η : π , → π , . The element h Q is ahidden value of τ h H under this map. This hidden value has target indeterminacygenerated by τ h X = h · ( τ X + τ C ′ ). Let α be an element of π a,b . Then multiplicationby α induces a filtration preserving map π p,q → π p + a,q + b . A hidden value of thismap is precisely the same as a hidden extension by α in the sense of [ , Definition4.2]. For clarity, we repeat the definition here. Definition . Let α be an element of π ∗ , ∗ that is detected by an element a of the E ∞ -page of the C -motivic Adams spectral sequence. A hidden extensionby α is a pair of elements b and c of E ∞ such that:(1) ab = 0 in the E ∞ -page.(2) There exists an element β of { b } such that αβ is contained in { c } .(3) If there exists an element β ′ of { b ′ } such that αβ ′ is contained in { c } , thenthe Adams filtration of b ′ is less than or equal to the Adams filtration of b .A crossing value for the map α : π p,q → π p + a,q + b is precisely the same as acrossing extension in the sense of [ , Examples 4.6 and 4.7].The discussion target indeterminacy applies to the case of hidden extensions.For example, the hidden η extension from h Q to τ h H has target indeterminacygenerated by τ h X .Typically, there is a symmetry in the presence of hidden extensions, in thefollowing sense. Let α and β be detected by a and b respectively. If there is ahidden α extension from b to c , then usually there is also a hidden β extension from a to c as well. However, this symmetry does not always occur, as the followingexample demonstrates. Example . We prove in Lemma 7.156 that ( σ + κ ) θ is zero. Therefore,there is no ( σ + κ ) extension on h . On the other hand, Table 21 shows that σ θ is non-zero and detected by h h A . Therefore, there is a hidden θ extension from h to h h A .In later chapters, we will thoroughly explore hidden extensions by 2, η , and ν .We warn the reader that a complete understanding of such hidden extensions doesnot necessarily lead to a complete understanding of multiplication by 2, η , and ν in the C -motivic stable homotopy groups. For example, in the 45-stem, there exists an element θ . that is detected by h h such that 4 θ . is detected by h h d . This is an example of a hidden 4 ex-tension. However, there is no hidden 2 extension from h h h to h h d ; condition(3) of Definition 2.6 is not satisfied.In fact, a complete understanding of all hidden extensions leads to a completeunderstanding of the multiplicative structure of the C -motivic stable homotopygroups, but the process is perhaps more complicated than expected.For example, we mentioned in Example 2.7 that either η ( σ + κ ) or ησ is non-zero, but these cases cannot be distinguished by a study of hidden η extensions.However, we can express that ησ is zero by observing that there is no hidden σ extension from h h to h d .There are even further complications. For example, the equation h + h h = 0does not prove that ν + η σ equals zero because it could be detected in higherfiltration. In fact, this does occur. Toda’s relation [ ] says that η σ + ν = ηǫ, where ηǫ is detected by h c .We can express Toda’s relation in terms of a “matric hidden extension”. Wehave a map [ ν η ] : π , ⊕ π , → π , . The associated graded map takes ( h , h h )to zero, but h c is the hidden value of ( h , h h ) under this map, in the sense ofDefinition 2.6. Over C , a “motivic modular forms” spectrum mmf has recently been con-structed [ ]. From our computational perspective, mmf is a ring spectrum whosecohomology is A//A (2), i.e., the quotient of the C -motivic Steenrod algebra by thesubalgebra generated by Sq , Sq , and Sq . By the usual change-of-rings isomor-phism, this implies that the homotopy groups of mmf are computed by an Adamsspectral sequence whose E -page is the cohomology of C -motivic A (2) [ ]. TheAdams spectral sequence for mmf has been completely computed [ ].By naturality, the unit map S , → mmf yields a map of Adams spectralsequences. This map allows us to transport information from the thoroughly un-derstood spectral sequence for mmf to the less well understood spectral sequencefor S , . This comparison technique is essential at many points throughout ourcomputations.We rely on notation from [ ] and [ ] for the Adams spectral sequence for mmf , except that we use a and n instead of α and ν respectively.For the most part, the map π ∗ , ∗ → π ∗ , ∗ mmf is detected on Adams E ∞ -pages.However, this map does have some hidden values. Theorem . Through dimension 90, Table 2 lists all hidden values of themap π ∗ , ∗ → π ∗ , ∗ mmf. Proof.
Most of these hidden values follow from hidden τ extensions in theAdams spectral sequences for S , and for mmf . For example, for S , , there is ahidden τ extension from h h g to d . For mmf , there is a hidden τ extension from cg to d . This implies that cg is the hidden value of h h g .A few cases are slightly more difficult. The hidden values of ∆ h h and h h i follow from the Adams-Novikov spectral sequences for S , and for mmf . Thesetwo values are detected on Adams-Novikov E ∞ -pages in filtration 2. .4. TODA BRACKETS 17 Next, the hidden value on
P h h j follows from multiplying the hidden value on h h i by d . Finally, the hidden values on ∆ h h , h h h i , and P h j follow fromalready established hidden values, relying on h extensions and h extensions. (cid:3) Remark . Through the 90-stem, there are no crossing values for the map π ∗ , ∗ → π ∗ , ∗ mmf . Moreover, in this range, there is only one hidden value that hastarget indeterminacy. Namely, ∆ h d is the hidden value of P h j , with targetindeterminacy generated by τ ∆ h g . C -motivic Steenrod algebra We have implemented machine computations of Ext, i.e., the cohomology ofthe C -motivic Steenrod algebra, in detail through the 110-stem. We take thiscomputational data for granted. It is depicted graphically in the chart of the E -page shown in [ ]. The data is available at [ ]. See [ ] for a discussion of theimplementation.In addition to the additive structure of Ext, we also have complete informationabout multiplications by h , h , h , and h . We do not have complete multiplicativeinformation. Occasionally we must deduce some multiplicative information on anad hoc basis.Similarly, we do not have systematic machine-generated Massey product infor-mation about Ext. We deduce some of the necessary information about Masseyproducts in Chapter 4.In the classical situation, Bruner has carried out extensive machine computa-tions of the cohomology of the classical Steenrod algebra [ ]. This data includescomplete primary multiplicative information, but no higher Massey product struc-ture. We rely heavily on this information. Our reliance on this data is so ubiquitousthat we will not give repeated citations.The May spectral sequence is the key tool for a conceptual computation ofExt. See [ ] for full details. In this manuscript, we use the May spectral sequenceto compute some Massey products that we need for various specific purposes; seeRemark 2.26 for more details.For convenience, we restate the following structural theorem about a portionof Ext C [ , Theorem 2.19]. Theorem . There is a highly structured isomorphism from
Ext cl to thesubalgebra of Ext consisting of elements in degrees ( s, f, w ) with s + f − w = 0 .This isomorphism takes classical elements of degree ( s, f ) to motivic elements ofdegree (2 s + f, f, s + f ) . Toda brackets are an essential computational tool for understanding stablehomotopy groups. Brackets appear throughout the various stages of the compu-tations, including in the analysis of Adams differentials and in the resolution ofhidden extensions.It is well-known that the stable homotopy groups form a ring under the com-position product. The higher Toda bracket structure is an extension of this ringstructure that is much deeper and more intricate. Our philosophy is that the sta-ble homotopy groups are not really understood until the Toda bracket structure isrevealed.
A complete analysis of all Toda brackets (even in a range) is not a practicalgoal. There are simply too many possibilities to take into account methodically,especially when including matric Toda brackets (and possibly other more exoticnon-linear types of brackets). In practice, we compute only the Toda brackets thatwe need for our specific computational purposes.
We next discuss the Moss Con-vergence Theorem [ ], which is the essential tool for computing Toda brackets instable homotopy groups. In order to make this precise, we must clarify the varioustypes of bracket operations that arise.First, the Adams E -page has Massey products arising from the fact that itis the homology of the cobar complex, which is a differential graded algebra. Wetypically refer to these simply as “Massey products”, although we write “Masseyproducts in the E -page” for clarification when necessary.Next, each higher E r -page also has Massey products, since it is the homologyof the E r − -page, which is a differential graded algebra. We always refer to theseas “Massey products in the E r -page” to avoid confusion with the more familiarMassey products in the E -page. This type of bracket appears only occasionallythroughout the manuscript.Beware that the higher E r -pages do not inherit Massey products from thepreceding pages. For example, τ h equals the Massey product h h , h , h i in the E -page. However, in the E -page, the bracket h h , h , h i equals zero, since theproduct h h is already equal to zero in the E -page before taking homology toobtain the E -page.On the other hand, the Massey product h h , h , h i is not a well-defined Masseyproduct in the E -page since h h is non-zero, while h h , h , h i in the E -pageequals h h because of the differential d ( h ) = h h .Finally, we have Toda brackets in the stable homotopy groups π ∗ , ∗ . The pointof the Moss Convergence Theorem is to relate these various kinds of brackets. Definition . Let a and b be elements in the E r -page of the C -motivicAdams spectral sequence such that ab = 0, and let n ≥
1. We call a nonzerodifferential d r + n x = y a crossing differential for the product ab in the E r page, if the element y has thesame stem and motivic weight as the product ab = 0, and the difference betweenthe Adams filtration of y and of ab is strictly greater than 0 but strictly less than n + 1.Figure 2 depicts the situation of a crossing differential in a chart for the E r -page. Typically, the product ab is zero in the E r -page because it was hit by a d r − differential, as shown by the dashed arrow in the figure. However, it may very wellbe the case that the product ab is already zero in the E r − -page (or even in anearlier page), in which case the dashed d r − differential is actually d r − (0) = 0. Theorem . Suppose that a , b , and c arepermanent cycles in the E r -page of the C -motivic Adams spectral sequence thatdetect homotopy classes α , β , and γ in π ∗ , ∗ respectively. Suppose further that(1) the Massey product h a, b, c i is defined in the E r -page, i.e., ab = 0 and bc = 0 in the E r -page. .4. TODA BRACKETS 19 Figure 2.
Crossing differentials xd r − aby d r + n (2) the Toda bracket h α, β, γ i is defined in π ∗ , ∗ , i.e., αβ = 0 and βγ = 0 .(3) there are no crossing differentials for the products ab and bc in the E r -page.Then there exists an element e contained in the Massey product h a, b, c i in the E r -page, such that(1) the element e is a permanent cycle.(2) the element e detects a homotopy class in the Toda bracket h α, β, γ i . Remark . The homotopy classes α , β , and γ are usually not unique. Thepresence of elements in higher Adams filtration implies that a , b , and c detect morethan one homotopy class. Moreover, it may be the case that h α, β, γ i is defined foronly some choices of α , β , and γ , while the Toda bracket is not defined for otherchoices. Remark . The Moss Convergence Theorem 2.16 says that a certain Masseyproduct h a, b, c i in the E r -page contains an element with certain properties. Thetheorem does not claim that every element of h a, b, c i has these properties. In thepresence of indeterminacies, there can be elements in h a, b, c i that do not satisfythe given properties. Remark . Beware that the Toda bracket h α, β, γ i may have non-zero in-determinacy. In this case, we only know that e detects one element of the Todabracket. Other elements of the Toda bracket could possibly be detected by otherelements of the Adams E ∞ -page; these occurrences must be determined by inspec-tion. Remark . In practice, one computes a Toda bracket h α, β, γ i by firststudying its corresponding Massey product h a, b, c i in a certain page of the Adamsspectral sequence. In the case that the Massey product h a, b, c i equals zero in the E r -page in Adams filtration f , the Moss Convergence Theorem 2.16 does not implythat the Toda bracket h α, β, γ i contains zero. Rather, the Toda bracket containsan element (possibly zero) that is detected in Adams filtration at least f + 1. Example . Consider the Toda bracket h ν, η, ν i . The elements h and h are permanent cycles that detect η and ν , and the product ην is zero. We havethat h h , h , h i equals h h , with no indeterminacy, in the E -page. There areno crossing differentials for the product h h = 0 in the E -page, so the MossConvergence Theorem 2.16 implies that h h detects a homotopy class in h ν, η, ν i . Note that h h detects the homotopy class ησ because h is a permanent cyclethat detects σ . However, we cannot conclude that ησ is contained in h ν, η, ν i . Thepresence of the permanent cycle c in higher filtration means that h h detectsboth ησ and ησ + ǫ , where ǫ is the unique homotopy class that is detected by c .The Moss Convergence Theorem 2.16 implies that either ησ or ησ + ǫ is containedin the Toda bracket h ν, η, ν i . In fact, ησ + ǫ is contained in the Toda bracket, butdetermining this requires further analysis. Example . Consider the Toda bracket h σ , , η i . The elements h , h , and h are permanent cycles that detect σ , 2, and η respectively, and the products 2 σ and 2 η are both zero. Due to the Adams differential d ( h ) = h h , the Masseyproduct h h , h , h i equals h h in the E -page, with zero indeterminacy. Thereare no crossing differentials for the products h h = 0 and h h = 0 in the E -page.The Moss Convergence Theorem 2.16 implies that h h detects a homotopy classin the Toda bracket h σ , , η i .The element h also detects σ + κ , where κ is the unique homotopy classthat is detected by d , and the product 2( σ + κ ) is zero. The Moss ConvergenceTheorem 2.16 also implies that h h detects a homotopy class in the Toda bracket h σ + κ, , η i . Example . Consider the Toda bracket h κ, , η i . The elements d , h , and h are permanent cycles that detect κ , 2, and η respectively, and the products 2 κ and 2 η are both zero. Due to the Adams differential d ( h h ) = h d , the Masseyproduct h d , h , h i equals h h · h = 0 in Adams filtration 3 in the E -page, withzero indeterminacy. There are no crossing differentials for the products h d = 0and h h = 0 in the E -page. The Moss Convergence Theorem 2.16 implies thatthe Toda bracket h κ, , η i either contains zero, or it contains a non-zero elementdetected in Adams filtration greater than 3.The only possible detecting element is P c . There is a hidden η extensionfrom h h to P c , so P c detects an element in the indeterminacy of h κ, , η i .Consequently, the Toda bracket is { , ηρ } , where ρ is detected by h h . Example . The Massey product h h , h , h i equals { f , f + h h h } inthe E -page. The elements h , h , and h are permanent cycles that detect ν , σ ,and 4 respectively, and the products νσ and 4 σ are both zero. However, theproduct h h has a crossing differential d ( h h ) = h d . The Moss ConvergenceTheorem 2.16 does not apply, and we cannot conclude anything about the Todabracket h ν, σ , i . In particular, we cannot conclude that { f , f + h h h } containsa permanent cycle. In fact, both elements support Adams d differentials. Remark . There is a version of the Moss Convergence Theorem 2.16 forcomputing fourfold Toda brackets h α, β, γ, δ i in terms of fourfold Massey products h a, b, c, d i in the E r -page. In this case, the crossing differential condition appliesnot only to the products ab , bc , and cd , but also to the subbrackets h a, b, c i and h b, c, d i . Remark . Just as the Moss Convergence Theorem 2.16 is the key tool forcomputing Toda brackets with the Adams spectral sequence, the May ConvergenceTheorem is the key tool for computing Massey products with the May spectralsequence. The statement of the May Convergence Theorem is entirely analogous tothe statement of the Moss Convergence Theorem, with Adams differentials replaced .4. TODA BRACKETS 21 by May differentials; Adams E r -pages replaced by May E r -pages; π ∗ , ∗ replacedby Ext; and Toda brackets replaced by Massey products. An analogous crossingdifferential condition applies. See [ , Section 2.2] [ ] for more details. We willuse the May Convergence Theorem to compute various Massey products that weneed for specific purposes. Occasionally, we will use Moss’s higherLeibniz rule [ ], which describes how Massey products in the E r -page interactwith the Adams d r differential. This theorem is a direct generalization of the usualLeibniz rule d r ( ab ) = d r ( a ) b + ad r ( b ) for twofold products. Theorem . [ ] Suppose that a , b , and c are elements in the E r -page ofthe C -motivic Adams spectral sequence such that ab = 0 , bc = 0 , d r ( b ) a = 0 , and d r ( b ) c = 0 . Then d r h a, b, c i ⊆ h d r ( a ) , b, c i + h a, d r ( b ) , c i + h a, b, d r ( c ) i , where all brackets are computed in the E r -page. Remark . By the Leibniz rule, the conditions d r ( b ) a = 0 and d r ( b ) c = 0imply that d r ( a ) b = 0 and d r ( c ) b = 0. Therefore, all of the Massey products inTheorem 2.27 are well-defined. Remark . The Massey products in Moss’s higher Leibniz rule 2.27 mayhave indeterminacies, so the statement involves an inclusion of sets, rather than anequality.
Remark . Beware that Moss’s higher Leibniz rule 2.27 cannot be appliedto Massey products in the E r -page to study differentials in higher pages. Forexample, we cannot use it to compute the d differential on a Massey product inthe E -page. In fact, there are versions of the higher Leibniz rule that apply tohigher differentials [ , Theorem 8.2] [ , Theorem 4.3], but these results havestrong vanishing conditions that often do not hold in practice. Example . Consider the element τ ∆ h , which was called G in [ ]. Table4 shows that there is an Adams differential d ( τ ∆ h ) = M h h , which follows bycomparison to Cτ . To illustrate Moss’s higher Leibniz rule 2.27, we shall give anindependent derivation of this differential.Table 3 shows that τ ∆ h equals the Massey product h h , h , D i , with no in-determinacy. By Moss’s higher Leibiz rule 2.27, the element d ( τ ∆ h ) is containedin h , h , D i + h h , , D i + h h , h , d ( D ) i . By inspection, the first two terms vanish. Also, Table 4 shows that d ( D ) equals h h g .Therefore, d ( τ ∆ h ) is contained in the bracket h h , h , h h g i , which equals h h , h , h g i h . Finally, Table 3 shows that h h , h , h g i equals M h . This showsthat d ( τ ∆ h ) equals M h h . Example . Consider the element τ e g in the Adams E -page. Because ofthe Adams differential d ( e ) = h d , we have that τ e g equals h d , h , τ g i in theAdams E -page. The higher Leibniz rule 2.27 implies that d ( τ e g ) is contained in h , h , τ g i + h d , , τ g i + h d , h , i , which equals { , c d } . In this case, the higher Leibniz rule 2.27 does not help todetermine the value of d ( τ e g ) because the indeterminacy is too large. (In fact, d ( τ e g ) does equal c d , but we need a different argument.) Example . Lemma 5.29 shows that d (∆ h h ) equals h h d . This argu-ment uses that ∆ h h equals h ∆ h , h , h i in the E -page, because of the Adamsdifferential d ( h ) = h h . Toda brackets satisfy varioustypes of formal relations that we will use extensively. The most important exampleof such a relation is the shuffle formula α h β, γ, δ i = h α, β, γ i δ, which holds whenever both Toda brackets are defined. Note the equality of setshere; the indeterminacies of both expressions are the same.The following theorem states some formal properties of threefold Toda bracketsthat we will use later. We apply these results so frequently that we typically usethem without further mention. Theorem . Let α , α ′ , β , γ , and δ be homotopy classes in π ∗ , ∗ . Each of thefollowing relations involving threefold Toda brackets holds up to a sign, wheneverthe Toda brackets are defined:(1) h α + α ′ , β, γ i ⊆ h α, β, γ i + h α ′ , β, γ i . (2) h α, β, γ i = h γ, β, α i . (3) α h β, γ, δ i ⊆ h αβ, γ, δ i . (4) h αβ, γ, δ i ⊆ h α, βγ, δ i . (5) α h β, γ, δ i = h α, β, γ i δ. (6) ∈ h α, β, γ i + h β, γ, α i + h γ, α, β i . We next turn our attention to fourfold Toda brackets. New complicationsarise in this context. If αβ = 0, βγ = 0, γδ = 0, h α, β, γ i contains zero, and h β, γ, δ i contains zero, then the fourfold bracket h α, β, γ, δ i is not necessarily defined.Problems can arise when both threefold subbrackets have indeterminacy. See [ ]for a careful analysis of this problem in the analogous context of Massey products.However, when at least one of the threefold subbrackets is strictly zero, thenthese difficulties vanish. Every fourfold bracket that we use has at least one threefoldsubbracket that is strictly zero.Another complication with fourfold Toda brackets lies in the description ofthe indeterminacy. If at least one threefold subbracket is strictly zero, then theindeterminacy of h α, β, γ, δ i is the linear span of the sets h α, β, ǫ i , h α, ǫ, δ i , and h ǫ, γ, δ i , where ǫ ranges over all possible values in the appropriate degree for whichthe Toda bracket is defined.The following theorem states some formal properties of fourfold Toda bracketsthat we will use later. We apply these results so frequently that we typically usethem without further mention. Theorem . Let α , α ′ , β , γ , δ , and ǫ be homotopy classes in π ∗ , ∗ . Each ofthe following relations involving fourfold Toda brackets holds up to a sign, wheneverthe Toda brackets are defined:(1) h α + α ′ , β, γ, δ i ⊆ h α, β, γ, δ i + h α ′ , β, γ, δ i . (2) h α, β, γ, δ i = h δ, γ, β, α i . .4. TODA BRACKETS 23 (3) α h β, γ, δ, ǫ i ⊆ h αβ, γ, δ, ǫ i . (4) h αβ, γ, δ, ǫ i ⊆ h α, βγ, δ, ǫ i . (5) α h β, γ, δ, ǫ i = h α, β, γ, δ i ǫ. (6) α h β, γ, δ, ǫ i ⊆ hh α, β, γ i , δ, ǫ i . (7) ∈ hh α, β, γ i , δ, ǫ i + h α, h β, γ, δ i , ǫ i + h α, β, h γ, δ, ǫ ii . Part (6) of Theorem 2.35 requires some further explanation. In the expres-sion hh α, β, γ i , δ, ǫ i , we have a set h α, β, γ i as the first input to a threefold Todabracket. The expression hh α, β, γ i , δ, ǫ i is defined to be the union of all sets of theform h ζ, δ, ǫ i , where ζ ranges over all elements of h α, β, γ i . Part (7) uses the samenotational convention.We will make occasional use of matric Toda brackets. We will not describe theirshuffling properties in detail, except to observe that they obey analogous matricversions of the properties in Theorems 2.34 and 2.35.HAPTER 3 The algebraic Novikov spectral sequence
Consider the cofiber sequence S , − τ / / S , i / / Cτ p / / S , − , where Cτ is the cofiber of τ . The inclusion i of the bottom cell and projection p to the top cell are tools for comparing the C -motivic Adams spectral sequencefor S , to the C -motivic Adams spectral sequence for Cτ . In [ ], we analyzedboth spectral sequences simultaneously, playing the structure of each against theother in order to obtain more detailed information about both. Then we used thestructure of the homotopy of Cτ to reverse-engineer the structure of the classicalAdams-Novikov spectral sequence.In this manuscript, we use Cτ in a different, more powerful way, because wehave a deeper understanding of the connection between the homotopy of Cτ andthe structure of the classical Adams-Novikov spectral sequence. Namely, the C -motivic spectrum Cτ is an E ∞ -ring spectrum [ ] (here E ∞ is used in the naivesense, not in some enriched motivic sense). Moreover, with appropriate finitenessconditions, the homotopy category of Cτ -modules is equivalent to the category of BP ∗ BP -comodules [ ] [ ]. By considering endomorphisms of unit objects, thiscomparison of homotopy categories gives a structured explanation for the identifi-cation of the homotopy of Cτ and the classical Adams-Novikov E -page.From a computational perspective, there is an even better connection. Namely,the algebraic Novikov spectral sequence for computing the Adams-Novikov E -page[ ] [ ] is identical to the C -motivic Adams spectral sequence for computing thehomotopy of Cτ . This rather shocking, and incredibly powerful, identification ofspectral sequences allows us to transform purely algebraic computations directlyinto information about Adams differentials for Cτ . Finally, naturality along theinclusion i of the bottom cell and along the projection p to the top cell allows usto deduce information about Adams differentials for S , .Due to the large quantity of data, we do not explicitly describe the structureof the Adams spectral sequence for Cτ in this manuscript. We refer the interestedreader to the charts in [ ], which provide details in a graphical form. Our naming convention for elements of the algebraic Novikov spectral sequence(and for elements of the Adams-Novikov spectral sequence) differs from previousapproaches. Our names are chosen to respect the inclusion i of the bottom cell andthe projection p to the top cell. Specifically, if x is an element of the C -motivicAdams E -page for S , , then we use the same letter x to indicate its image i ∗ ( x )in the Adams E -page for Cτ . It is certainly possible that i ∗ ( x ) is zero, but we
256 3. THE ALGEBRAIC NOVIKOV SPECTRAL SEQUENCE will only use this convention in cases where i ∗ ( x ) is non-zero, i.e., when x is not amultiple of τ .On the other hand, if x is an element of the C -motivic Adams E -page for S , such that τ x is zero, then we use the symbol x to indicate an element of p − ∗ ( x )in the Adams E -page for Cτ . There is often more than one possible choice for x , and the indeterminacy in this choice equals the image of i ∗ in the appropriatedegree. We will not usually be explicit about these choices. However, potentialconfusion can arise in this context. For example, it may be the case that one choiceof x supports an h extension, while another choice of x supports an h extension,but there is no possible choice of x that simultaneously supports both extensions.(The authors dwell on this point because this precise issue has generated confusionabout specific computations.) We have analyzed the algebraic Novikov spectral sequence by computer in alarge range. Roughly speaking, our algorithm computes a Curtis table for a minimalresolution. Significant effort went into optimizing the linear algebra algorithms tocomplete the computation in a reasonable amount of time. The data is availableat [ ]. See [ ] for a discussion of the implementation.Our machine computations give us a full description of the additive structure ofthe algebraic Novikov E -page, together with all d r differentials for r ≥
2. It thusyields a full description of the additive structure of the algebraic Novikov E ∞ -page.Moreover, the data also gives full information about multiplication by 2, h ,and h in the Adams-Novikov E -page for the classical sphere spectrum, which wedenote by H ∗ ( S ; BP ).We have also conducted machine computations of the Adams-Novikov E -page for the classical cofiber of 2, which we denote by H ∗ ( S/ BP ). Note that H ∗ ( S ; BP ) is the homology of a differential graded algebra (i.e., the cobar com-plex) that is free as a Z -module. Therefore, H ∗ ( S/ BP ) is the homology of thisdifferential graded algebra modulo 2. We have computed this homology by ma-chine, including full information about multiplication by h , h , and h . Thesecomputations are related by a long exact sequence · · · / / H ∗ ( S ; BP ) j / / H ∗ ( S/ BP ) q / / H ∗ ( S ; BP ) / / · · · Because h , h , h , and h are annihilated by 2 in H ∗ ( S ; BP ), there are classes f h , f h , f h , and f h in H ∗ ( S/ BP ) such that q ( f h i ) equals h i for 2 ≤ i ≤
5. We alsohave full information about multiplication by f h , f h , f h , and f h in H ∗ ( S/ BP )This multiplicative information allows use to determine some of the Masseyproduct structure in the Adams-Novikov E -page for the sphere spectrum. Thereare several cases to consider.First, let x and y be elements of H ∗ ( S ; BP ). If the product j ( x ) j ( y ) is non-zeroin H ∗ ( S/ BP ), then xy must also be non-zero in H ∗ ( S ; BP ).In the second case, let x be an element of H ∗ ( S ; BP ), and let e y be an elementof H ∗ ( S/ BP ) such that q ( e y ) = y . If the product x · e y is non-zero in H ∗ ( S/ BP )and equals j ( z ) for some z in H ∗ ( S ; BP ), then z belongs to the Massey product h , y, x i . This follows immediately from the relationship between Massey productsand the multiplicative structure of a cofiber, as discussed in [ , Section 3.1.1]. .3. h -BOCKSTEIN SPECTRAL SEQUENCE 27 Third, let e x and e y be elements of H ∗ ( S/ BP ) such that q ( e x ) = x , q ( e y ) = y ,and q ( e x · e y ) = z . Then z belongs to the Massey product h x, , y i in H ∗ ( S ; BP ).This follows immediately from the multiplicative snake lemma 3.3. Example . Computer data shows that the product f h · f h does not equalzero in H ∗ ( S/ BP ). This implies that the Massey product h h , , h i does notcontain zero in H ∗ ( S ; BP ), which in turn implies that the Toda bracket h θ , , θ i does not contain zero in π , . Remark . let e x and e y be elements of H ∗ ( S/ BP ) such that q ( e x ) and q ( e y )equal x and y , and such that e x · e y equals j ( z ) for some z in H ∗ ( S ; BP ). It appearsthat z has some relationship to the fourfold Massey product h , x, , y i , but we havenot made this precise. Lemma . Let A be a differential graded al-gebra that has no -torsion, and let H ( A ) be its homology. Also let H ( A/ be thehomology of A/ , and let δ : H ( A/ → H ( A ) be the boundary map associated tothe short exact sequence / / A / / A / / A/ / / . Suppose that a and b are elements of H ( A/ such that δ ( a ) = 0 and δ ( b ) = 0 in H ( A ) . Then the Massey product h δ ( a ) , , δ ( b ) i in H ( A ) contains δ ( ab ) . Proof.
We carry out a diagram chase in the spirit of the snake lemma. Write ∂ for the boundary operators in A and A/ x and y be cycles in A/ a and b respectively. Let x ′ and y ′ be elements in A that reduce to x and y . Then ∂x ′ and ∂y ′ reduce to zero in A/ x and y are cycles. Therefore, ∂x ′ = 2 e x and ∂y ′ = 2 e y for some e x and e y in A . By definition of the boundary map, δ ( a ) and δ ( b ) are represented by e x and e y . Bythe definition of Massey products, the cycle e xy ′ + x ′ e y is contained in h δ ( a ) , , δ ( b ) i .Now we compute δ ( ab ). Note that x ′ y ′ is an element of A that reduces to ab .Then ∂ ( x ′ y ′ ) = ∂ ( x ′ ) y ′ + x ′ ∂ ( y ′ ) = 2( e xy ′ + x ′ e y ) . This shows that δ ( ab ) is represented by e xy ′ + x ′ e y . (cid:3) h -Bockstein spectral sequence The charts in [ ] show graphically the algebraic Novikov spectral sequence,i.e., the Adams spectral sequence for Cτ . Essentially all of the information inthe charts can be read off from machine-generated data. This includes hiddenextensions in the E ∞ -page.One aspect of these charts requires further explanation. The C -motivic Adams E -page for Cτ contains a large number of h -periodic elements, i.e., elementsthat support infinitely many h multiplications. The behavior of these elementsis entirely understood [ ], at least up to many multiplications by h , i.e., in an h -periodic sense.On the other hand, it takes some work to “desperiodicize” this information.For example, we can immediately deduce from [ ] that d ( h k e ) = h k +21 d forlarge values of k , but that does not necessarily determine the behavior of Adamsdifferentials for small values of k . The behavior of these elements is a bit subtle in another sense, as illustratedby Example 3.4.
Example . Consider the h -periodic element c e . Machine computationstell us that this element supports a d differential, but there is more than onepossible value for d ( c e ) because of the presence of both h c d and P e .In fact, d ( c d ) equals P d , and d ( P e ) equals P h d . Therefore, P e + h c d is the only non-zero d cycle, and it follows that d ( c e ) must equal P e + h c d .In higher stems, it becomes more and more difficult to determine the exactvalues of the Adams d differentials on h -periodic classes. Eventually, these com-plications become unmanageable because they involve sums of many monomials.Fortunately, we only need concern ourselves with the Adams d differential inthis context. The h -periodic E -page equals the h -periodic E ∞ -page, and theonly non-zero classes are well-understood v -periodic families running along thetop of the Adams chart.Our solution to this problem, as usual, is to introduce a filtration that hides thefiltration. In this case, we filter by powers of h . The effect is that terms involvinghigher powers of h are ignored, and the formulas become much more manageable.This h -Bockstein spectral sequence starts with an E -page, because there aresome differentials that do not increase h divisibility. For example, we have Bock-stein differentials d ( h e ) = h d and d ( c d ) = P d , reflecting the Adams dif-ferentials d ( h e ) = h d and d ( c d ) = P d .There are also plenty of higher h -Bockstein differentials, such as d ( e ) = h d ,and d ( e g ) = M h . Remark . Beware that filtering by powers of h changes the multiplica-tive structure in perhaps unexpected ways. For example, P h and d are not h -multiples, so their h -Bockstein filtration is zero. One might expect their prod-uct to be P h d , but the h -Bockstein filtration of this element is 1. Therefore, P h · d equals 0 in the h -Bockstein spectral sequence.But not all P h multiplications are trivial in the h -Bockstein spectral sequence.For example, we have P h · c d = P h c d because the h -Bockstein filtrations ofall three elements are zero.In Example 3.4, we explained that there is an Adams differential d ( c e ) = P e + h c d . When we throw out higher powers of h , we obtain the h -Bocksteindifferential d ( c e ) = P e . We also have an h -Bockstein differential d ( c d ) = P d .The first four charts in [ ] show graphically how this h -Bockstein spectralsequence plays out in practice. The main point is that the h -Bockstein E ∞ -pagereveals which (formerly) h -periodic classes contribute to the Adams E -page for Cτ .HAPTER 4 Massey products
The purpose of this chapter is to provide some general tools, and to give somespecific computations, of Massey products in Ext. This material contributes toTable 3, which lists a number of Massey products in Ext that we need for variousspecific purposes. Most commonly, these Massey products yield information aboutToda brackets via the Moss Convergence Theorem 2.16.We begin with a C -motivic version of a classical theorem of Adams aboutsymmetric Massey products [ , Lemma 2.5.4]. Theorem . (1) If h x is zero, then h h , x, h i contains τ h x .(2) If n ≥ and h n x is zero, then h h n , x, h n i contains h n +1 x . g The projection map p : A ∗ → A (2) ∗ induces a map p ∗ : Ext C → Ext A (2) .Because Ext A (2) is completely known [ ], this map is useful for detecting structurein Ext C . Proposition 4.2 provides a tool for using p ∗ to compute certain types ofMassey products. Proposition . Let x be an element of Ext C such that h x = 0 . Then p ∗ (cid:0) h h , h , x i (cid:1) equals the element gp ∗ ( x ) in Ext A (2) . Proof.
The idea of the proof is essentially the same as in [ , Proposition 3.1].The Ext C -module Ext A (2) is a “Toda module”, in the sense that Massey products h x, a, b i are defined for all x in Ext A (2) and all a and b in Ext C such that x · a = 0and ab = 0. In particular, the bracket h , h , h i is defined in Ext A (2) . We wish tocompute this bracket.We use the May Convergence Theorem in order to compute the bracket. Thecrossing differentials condition on the theorem is satisfied because there are nopossible differentials that could interfere.The key point is the May differential d ( b ) = h h . This shows that g iscontained in h , h , h i . Also, the bracket has no indeterminacy by inspection.Now suppose that x is an element of Ext C such that h x = 0. Then p ∗ (cid:0) h h , h , x i (cid:1) = 1 · h h , h , x i = h , h , h i · x = gp ∗ ( x ) . (cid:3) Example . We illustrate the practical usefulness of Proposition 4.2 witha specific example. Consider the Massey product h h h , h , h i . The propositionsays that p ∗ (cid:0) h h h , h , h i (cid:1) = h g in Ext A (2) . This implies that h h h , h , h i equals h g in Ext C .
290 4. MASSEY PRODUCTS
Remark . The Massey product computation in Example 4.3 is in relativelylow dimension, and it can be computed using other more direct methods. Table3 lists additional examples, including some that cannot be determined by moreelementary methods.
We recall some results from [ ] about the Mahowald operator. The Mahowaldoperator is defined to be M x = h x, h , g i for all x such that h x equals zero. Asalways, one must be cautious about indeterminacy in M x .There exists a subalgebra B of the C -motivic Steenrod algebra whose coho-mology Ext B ( M , M ) equals M [ v ] ⊗ M Ext A (2) . The inclusion of B into the C -motivic Steenrod algebra induces a map p ∗ : Ext C → Ext B . Proposition . [ , Theorem 1.1] The map p ∗ : Ext C → Ext B takes M x tothe product ( e v + h v ) p ∗ ( x ) , whenever M x is defined.
Proposition 4.5 is useful in practice for detecting certain Massey products ofthe form h x, h , g i . For example, if x is an element of Ext C such that h x equalszero and e p ∗ ( x ) is non-zero in Ext A (2) , then h x, h , g i is non-zero. Example . Proposition 4.5 shows that h h , h , h g i is non-zero. Thereis only one non-zero element in the appropriate degree, so we have identified theMassey product. We give this element the name M h . Example . Expanding on Example 4.6, Proposition 4.5 also shows that h M h , h , h g i is non-zero. Again, there is only one non-zero element in the ap-propriate degree, so we have identified the Massey product. We give this elementthe name M h . Lemma . The Massey product h h h , h , τ gn i equals τ g n , with indetermi-nacy generated by M h h g . Proof.
We start by analyzing the indeterminacy. The product
M c · h h equals h g , h , c i h h = h g , h , h h c i = h g , h , h h · h g i = h g , h , h g i h h , which equals M h h g . The equalities hold because the indeterminacies are zero,and the first and last brackets in this computation are given by Table 3. This showsthat M h h g belongs to the indeterminacy.Table 3 shows that h h , h h , h i = h h , h , h h i equals h g . Then h h h h , h , τ gn i = h h , h h , h i τ gn = τ h g n. This implies that h h h , h , τ gn i contains either τ g n or τ g n + ∆ h g . However,the shuffle h h h h , h , τ gn i = h h , h h , h i τ gn = 0eliminates τ g n + ∆ h g . (cid:3) .3. ADDITIONAL COMPUTATIONS 31 Remark . The Massey product of Lemma 4.8 cannot be established withProposition 4.2 because p ∗ ( τ gn ) = 0 in Ext A (2) . Lemma . The Massey product h ∆ e + C , h , h h i equals (∆ e + C ) g ,with no indeterminacy. Proof.
Consider the Massey product h τ (∆ e + C ) , h , h i . By inspection,this Massey product has no indeterminacy. Therefore, h τ (∆ e + C ) , h , h i = (∆ e + C ) h τ, h , h i . Table 3 shows that the latter bracket equals τ g , so the expression equals τ (∆ e + C ) g .On the other hand, it also equals τ h ∆ e + C , h , h h i . Therefore, the bracket h ∆ e + C , h , h h i must contain (∆ e + C ) g . Finally, the indeterminacy can becomputed by inspection. (cid:3) Lemma . The Massey product h h , p ′ , h i equals h e , with no indetermi-nacy. Proof.
We have h h , p ′ , h i h = h h p ′ , h , h i = p ′ h h , h , h i . Table 3 shows that the last Massey product equals c . Observe that p ′ c equals h h e .This shows that h h , p ′ , h i equals either h e or h e + h e . However, shuffleto obtain h h , p ′ , h i h = h h p ′ , h , h i , which must equal zero. Since h ( h e + h e ) is non-zero, it cannot equal h h , p ′ , h i .The indeterminacy is zero by inspection. (cid:3) Lemma . The Massey product h τ gG , h h , h i has indeterminacy gener-ated by τ M h , and it either contains zero or τ e x , . In particular, it does notcontain any linear combination of ∆ h g with other elements. Proof.
The indeterminacy can be computed by inspection.The only possible elements in the Massey product h τ gG , h h , h i are linearcombinations of e x , and M h . The inclusion τ h τ gG , h h , h i ⊆ h τ gG , h h , h i gives the desired result. (cid:3) Lemma . The Massey product h h , h , h , h i equals ∆ h . Proof.
Table 3 shows that ∆ h equals the Massey product h h , h , h , h i .Recall the isomorphism between classical Ext groups and C -motivic Ext groups indegrees satisfying s + f − w = 0, as described in Theorem 2.14. This shows that∆ h equals h h , h , h , h i . (cid:3) HAPTER 5
Adams differentials
The goal of this chapter is to describe the values of the Adams differentialsin the motivic Adams spectral sequence. These values are given in Tables 4, 6, 7,8, and 9. See also the Adams charts in [ ] for a graphical representation of thecomputations. d differential Theorem . Table 4 lists the values of the Adams d differential on allmultiplicative generators, through the 95-stem, except that:(1) d ( D ′ ) equals either h X or h X + τ h C ′′ .(2) d ( x , ) might equal τ h x , .(3) d ( x , ) equals either x , or x , + τ d H . Proof.
The fourth column of Table 4 gives information on the proof of eachdifferential. Most follow immediately by comparison to the Adams spectral se-quence for Cτ . A few additional differentials follow by comparison to the classicalAdams spectral sequence for tmf .If an element is listed in the fourth column of Table 4, then the correspondingdifferential can be deduced from a straightforward argument using a multiplicativerelation. For example, it is possible that d (∆ h h ) equals τ d e . However, h · ∆ h h is zero, while h · τ d e is non-zero. Therefore, d (∆ h h ) must equal zero.In some cases, it is necessary to combine these different techniques to establishthe differential.The remaining more difficult computations are carried out in the followinglemmas. (cid:3) Table 4 lists all of the multiplicative generators of the Adams E -page throughthe 95-stem. The third column indicates the value of the d differential, if it isnon-zero. A blank entry in the third column indicates that the d differential iszero. The fourth column indicates the proof. A blank entry in the fourth columnindicates that there are no possible values for the differential. The fifth columngives alternative names for the element, as used in [ ], [ ], and [ ]. Remark . Note that d ( x , ) is non-zero in the Adams spectral sequencefor Cτ . Therefore, either d ( x , ) equals τ h x , , or d ( x , ) equals h x , . Ineither case, x , does not survive to the E ∞ -page, and either τ h x , or h x , is hit by a differential. Lemma . d (∆ x ) = h B + τ M h d . Proof.
We have a differential d (∆ x ) = h B in the Adams spectral sequencefor Cτ . Therefore, d (∆ x ) equals either h B or h B + τ M h d .
334 5. ADAMS DIFFERENTIALS
We have the relation h · ∆ x = P h · τ ∆ h , so h d (∆ x ) = P h d ( τ ∆ h ) = P h h · M h = τ M h d . Therefore, d (∆ x ) must equal h B + τ M h d . (cid:3) Remark . The proof of [ , Lemma 3.50] is incorrect. We claimed that h · ∆ x equals h · ∆ h h , when in fact h · ∆ x equals τ h · ∆ h h . Lemma . d ( x , ) = τ M h h . The following proof was suggested to us by Dexter Chua.
Proof.
This follows from the interaction between algebraic squaring opera-tions and classical Adams differentials [ , Theorem 2.2], applied to the element x in the 37-stem. The theorem says that d ∗ Sq x = Sq d x ∔ h Sq x. The notation means that there is an Adams differential on Sq x hitting eitherSq d x = 0 or h Sq x , depending on which element has lower Adams filtration.Therefore d Sq x = h Sq x .Next, observe from [ ] that Sq x = h x , + τ d g , so h Sq x = h x , = τ M h h . Therefore, there is a d differential whose value is τ M h h , and the possibility isthat d ( x , ) equals τ M h h . (cid:3) Lemma . d ( τ B g ) = τ M h g . Proof.
We use the Mahowald operator methods of Section 4.2. The map p ∗ : Ext C → Ext B takes d · τ B g to τ h ag v , which is non-zero. We deduce thatthe product d · τ B g is non-zero in Ext, and the only possibility is that it equals τ M g · h m .Now d ( τ M g · h m ) equals τ M g · h e , which we also know is non-zero sinceit maps to the non-zero element τ h deg v of Ext B . It follows that d ( τ B g ) mustequal τ M h g . (cid:3) d differential Theorem . Table 6 lists some values of the Adams d differential on mul-tiplicative generators. The Adams d differential is zero on all multiplicative gen-erators not listed in the table. The list is complete through the 95-stem, exceptthat: (1) d ( h h ) equals either h · ∆ h or h · ∆ h + h ( τ M e + h · ∆ x ) .(2) d (∆ h h ) equals either τ ∆ h e g or τ ∆ h e g + τ ∆ h d e .(3) d ( h h g ) might equal τ h h Q .(4) d (∆ h d ) equals either τ ∆ h d e or τ ∆ h d e + P ∆ h d e .(5) d ( x , ) might equal h x , .(6) d (∆ M h ) might equal τ M d e . Proof.
The d differential on many multiplicative generators is zero. A fewof these multiplicative generators appear in Table 6 because their proofs requirefurther explanation. For the remaining majority of such multiplicative generators,the d differential is zero because there are no possible non-zero values, becauseof comparison to the Adams spectral sequence for Cτ , or because the element is .2. THE ADAMS d DIFFERENTIAL 35 already known to be a permanent cycle as shown in Table 5. These cases do notappear in Table 6.The last column of Table 6 gives information on the proof of each differential.Most follow immediately by comparison to the Adams spectral sequence for Cτ .A few additional differentials follow by comparison to the classical Adams spectralsequence for tmf , or by comparison to the C -motivic Adams spectral sequence for mmf .If an element is listed in the last column of Table 6, then the correspondingdifferential can be deduced from a straightforward argument using a multiplicativerelation. For example, d ( h · τ P d e ) = P h · d ( τ d e ) = P h c d , so d ( τ P d e ) must equal P c d .If a d differential is listed in the last column of Table 6, then the correspondingdifferential is forced by consistency with that later differential. In each case, a d differential on an element x is forced by the existence of a later d differential on τ x .For example, Table 7 shows that there is a differential d ( τ e g ) = P d . Therefore, τ e g cannot survive to the E -page. It follows that d ( τ e g ) = c d .In some cases, it is necessary to combine these different techniques to establishthe differential.The remaining more difficult computations are carried out in the followinglemmas. (cid:3) Table 6 lists the multiplicative generators of the Adams E -page through the95-stem whose d differentials are non-zero, or whose d differentials are zero fornon-obvious reasons. Remark . Several of the uncertainties in the values of the d differentialare inconsequential because they do not affect the structure of later pages.(1) The uncertainty in d ( h h ) is inconsequential because d ( τ C ′ ) equals h ( τ M e + h · ∆ x ).(2) The uncertainty in d (∆ h h ) is inconsequential because d ( τ d B ) =∆ h d e .(3) The uncertainty in d (∆ h d ) is inconsequential since d ( τ M h d k ) = P ∆ h d e . Remark . Note that d ( x , ) is non-zero in the Adams spectral sequencefor Cτ . Therefore, either d ( x , ) equals τ h x , , or d ( x , ) equals h x , . Ineither case, x , does not survive to the E ∞ -page, and either τ h x , or h x , is hit by a differential. Remark . One other d differential possesses a different kind of uncer-tainty. We know that either τ D ′ or τ D ′ + τ h G survives to the E -page. InLemma 5.15, we show that the surviving element supports a d differential hitting τ M h g . This is an uncertainty in the source of the differential, rather than thetarget, which arises from an uncertainty about the value of d ( D ′ ). Proposition . Some permanent cycles in the C -motivic Adams spectralsequence are shown in Table 5. Proof.
The third column of the table gives information on the proof for eachelement. If a Toda bracket is given in the third column, then the Moss Convergence
Theorem 2.16 implies that the element must survive to detect that Toda bracket(see Table 11 for more information on how each Toda bracket is computed). If aproduct is given in the third column, then the element must survive to detect thatproduct (see Table 21 for more information on how each product is computed). Ina few cases, the third column refers to a specific lemma that gives a more detailedargument. (cid:3)
Lemma . (1) d ( h h ) = τ h d .(2) d ( P h h ) = τ h h Q . Proof.
In the Adams spectral sequence for Cτ , there is an η extension from h h to h d . The element h d maps to h d under projection from Cτ to the topcell, so h h must also map non-trivially under projection from Cτ to the top cell.The only possibility is that h h maps to h d . Therefore, τ h d must be hit by adifferential. This establishes the first differential.The proof for the second differential is identical, using that there is an η ex-tension from P h h to h h Q in the Adams spectral sequence for Cτ . (cid:3) Lemma . d ( τ ∆ h ) = τ M c . Proof.
The element
M P maps to zero under inclusion of the bottom cell into Cτ . Therefore, M P is either hit by a differential, or it is the target of a hidden τ extension. There are no possible differentials, so there must be a hidden τ extension.The only possibility is that τ M c is zero in the E ∞ -page, and that there is a hidden τ extension from M c to M P . (cid:3) Lemma . d ( τ h · ∆ g ) = τ ∆ h e g . Proof.
Table 2 shows that the element h h i maps to ∆ h in the Adamsspectral sequence for tmf .Now ∆ h d is not zero and not divisible by 2 in tmf . Therefore, κ { h h i } mustbe non-zero and not divisible by 2 in π , . The only possibility is that κ { h h i } is detected by P h h j = d · h h i , and that P h h j is not an h multiple in the E ∞ -page. Therefore, τ ∆ g · h cannot survive to the E ∞ -page. (cid:3) Lemma . Either d ( τ D ′ ) = τ M h g or d ( τ D ′ + τ h G ) = τ M h g ,depending only on whether τ D ′ or τ D ′ + τ h G survives to the E -page. Proof.
Table 11 shows that the Toda bracket h , σ, , σ i contains τ νκ , whichis detected by τ h g . Table 19 shows that M h detects να for some α in π , de-tected by h h . (Beware that there is a crossing extension, M h does not detect να for every α that is detected by h h .) It follows that τ M h g detects h , σ, , σ i α .This expression is contained in h , σ, h , σ , α ii . Lemma 6.12 shows that theinner bracket equals { , τ κ } .The Toda bracket h , σ, i in π , consists entirely of multiples of 2. TheToda bracket h , σ, τ κ i contains h , σ, i τ κ . This last expression equals equalszero because h , σ, i = τ η · σ = 0by Corollary 6.2. Therefore, h , σ, τ κ i equals its indeterminacy, which consistsentirely of multiples of 2 in π , . .2. THE ADAMS d DIFFERENTIAL 37
We conclude that τ M h g is either hit by a differential, or is the target of ahidden 2 extension. Lemma 7.24 shows that there is no hidden 2 extension from h A ′ to τ M h g , and there are no other possible extensions to τ M h g .Therefore, τ M h g must be hit by a differential, and the only possible sourcesof this differential are τ D ′ or τ D ′ + τ h G , depending on which element survivesto the E -page. (cid:3) Lemma . d ( τ M h l ) = ∆ h d . Proof.
Table 7 shows that d ( τ d e + h h ) equals P d , so d ( τ M h d e )equals τ M P h d . We have the relation h · τ M h l = τ M h d e , but the element τ M P h d is not divisible by h . Therefore, τ M h l cannot survive to the E -page.By comparison to the Adams spectral sequence for tmf , the value of d ( τ M h l )cannot be τ ∆ h e + ∆ h d or τ ∆ h e . The only remaining possibility is that d ( τ M h l ) equals ∆ h d . (cid:3) Lemma . d ( h x , ) = τ e g . Proof.
Suppose that h x , were a permanent cycle. Then it would mapunder inclusion of the bottom cell to the element h x , in the Adams E ∞ -pagefor Cτ .There is a hidden ν extension from h x , to ∆ h h in the Adams E ∞ -pagefor Cτ . Then ∆ h h would also have to be in the image of inclusion of the bottomcell. The only possible pre-image is the element ∆ h h in the Adams spectralsequence for the sphere, but this element does not survive by Lemma 5.44.By contradiction, we have shown that h x , must support a differential. Theonly possibility is that d ( h x , ) equals τ e g . (cid:3) Lemma . d ( x ) = τ h m . Proof.
This follows from the interaction between algebraic squaring opera-tions and classical Adams differentials [ , Theorem 2.2]. The theorem says that d ∗ Sq e = Sq d e ∔ h Sq e . The notation means that there is an Adams differential on Sq e hitting eitherSq d e or h Sq e , depending on which element has lower Adams filtration.Therefore d Sq e = h Sq e .Finally, we observe from [ ] that Sq e = x and Sq e = m . (cid:3) Lemma . The element ∆ d is a permanent cycle. Proof.
The element ∆ d in the Adams E ∞ -page for Cτ must map to zerounder the projection from Cτ to the top cell. The only possible value in sufficientlyhigh filtration is τ ∆ h e g . However, comparison to mmf shows that this elementis not annihilated by τ , and therefore cannot be in the image of projection to thetop cell.Therefore, ∆ d must be in the image of the inclusion of the bottom cell into Cτ . The element ∆ d is the only possible pre-image in the Adams E ∞ -page forthe sphere in sufficiently low filtration. (cid:3) Lemma . d (∆ h h ) equals either τ ∆ h e g or τ ∆ h e g + τ ∆ h d e . Proof.
There is a relation
P h · ∆ h h = τ ∆ h d in the Adams E -page.Because of the differential d (∆ h e ) = ∆ h d + τ e gm , we have the relation P h · ∆ h h = τ e gm in the E -page.There is a differential d ( τ e gm ) = τ d l . But τ d l is not divisible by P h ,so τ e gm cannot be divisible by P h in the E -page. Therefore, d (∆ h h ) mustbe non-zero.The same argument shows that d (∆ h h + τ d B ) must also be non-zero.Because of Lemma 5.21, the only possibilities are that d (∆ h h ) equals either τ ∆ h e g or τ ∆ h e g + τ ∆ h d e . (cid:3) Lemma . d ( τ d B ) = ∆ h d e . Proof.
The element ∆ h d e is a permanent cycle because there are no pos-sible differentials that it could support. Moreover, it must map to zero under theinclusion of the bottom cell into Cτ because there are no elements in the Adams E ∞ -page for Cτ of sufficiently high filtration. Therefore, ∆ h d e is either hit bya differential, or it is the target of a hidden τ extension.The only possible hidden τ extension has source h x , . However, Table 13shows that h x , is in the image of projection from Cτ to the top cell. Therefore,it cannot support a hidden extension.We now know that ∆ h d e must be hit by a differential. Lemma 5.19 rulesout one possible source for this differential. The only remaining possibility is that d ( τ d B ) equals d (∆ d ). (cid:3) Lemma . (1) d ( h h h ) = 0 .(2) d ( P h h ) = 0 . Proof.
The value of d ( h h h ) is not h h d nor h h d + τ h x by com-parison to the Adams spectral sequence for Cτ .It remains to show that d ( h h h ) cannot equal τ h x . Suppose that thedifferential did occur. Then there would be no possible targets for a hidden τ extension on h x , so the η extension from h x to h x would be detected byprojection from Cτ to the top cell. But there is no such η extension in the homotopygroups of Cτ . This establishes the first formula.The proof of the second formula is essentially the same, using that the η ex-tension from ∆ h d to ∆ h d cannot be detected by projection from Cτ to thetop cell. (cid:3) Lemma . d (∆ p ) = 0 . Proof.
Suppose that d (∆ p ) were equal to τ h M e . In the Adams E -page,the Massey product h τ M g, τ h d , d i would equal τ M ∆ h g , with no indetermi-nacy, because of the Adams differential d (∆ h ) = τ h d and because d · ∆ p = 0.By Moss’s higher Leibniz rule 2.27, d ( τ M ∆ h g ) would be a linear combination ofmultiples of τ M g and d . But Table 7 shows that d ( τ M ∆ h g ) equals M P ∆ h e ,which is not such a linear combination in the Adams E -page. (cid:3) Lemma . d ( τ h g + τ h e ) = 0 . Proof.
In the Adams E -page, we have the matric Massey product τ h g + τ h e = (cid:28)(cid:2) τ g τ h (cid:3) , (cid:20) h x (cid:21) , h (cid:29) .2. THE ADAMS d DIFFERENTIAL 39 because of the Adams differentials d ( h ) = h h and d ( e ) = h x , as well asthe relation τ g · h + τ h x in the Adams E -page. Moss’s higher Leibniz rule2.27implies that d ( τ h g + τ h e ) belongs to (cid:28) [0 0] , (cid:20) h x (cid:21) , h (cid:29) + (cid:28) [ τ g τ h ] , (cid:20) τ h m (cid:21) , h (cid:29) + (cid:28) [ τ g τ h ] , (cid:20) h x (cid:21) , (cid:29) since d ( x ) = τ h m , where the Massey products are formed in the Adams E -pageusing the d differential. This expression simplifies to (cid:28) [ τ g τ h ] , (cid:20) τ h m (cid:21) , h (cid:29) ,which equals { , τ h h Q } .Table 19 shows that there is a hidden ν extension from h h Q to P h x , .The element τ P h x , is non-zero in the Adams E ∞ -page. Therefore, h h Q supports a (hidden or not hidden) τ extension whose target is in Adams filtrationat most 10. The only possibility is that τ h h Q is non-zero in the Adams E ∞ -page. (cid:3) Lemma . d ( x , ) = 0 . Proof.
Suppose that d ( x , ) equaled τ h gD . Then h gD could not sup-port a hidden τ extension. The only possible target would be M ∆ h d , but that iseliminated by the hypothetical hidden τ extension on ∆ h j given in Remark 7.6.Table 19 shows that there is a hidden ν extension from h gD to B d . Thishidden extension would be detected by projection from Cτ to the top cell. Butthere is no such ν extension in the homotopy of Cτ . (cid:3) Lemma . d ( τ M h d k ) = P ∆ h d e . Proof.
Table 7 shows that d ( τ M h e ) = M P h . Multiply by τ d tosee that d ( τ M h d e ) = τ M P h d . We have the relation h · τ M h d k = τ M h d e , but τ M P h d is not divisible by h . Therefore, τ M h d k cannotsurvive to the E -page. By comparison to mmf , there is only one possible value for d ( τ M h d k ). (cid:3) Lemma . d ( h B g ) = M h c e . Proof.
First observe the relation d · h B g = τ M h e g . This relation fol-lows from [ , Theorem 1.1], modulo a possible error term P h h c e . However,multiplication by h eliminates this possibility.Table 6 shows that d ( τ e g ) = c d e . Therefore, d ( τ M h e g ) equals M h c d e . Observing that M h c d e is in fact non-zero in the Adams E -page,we conclude that d ( h B g ) must equal M h c e (cid:3) Lemma . The element M is a permanent cycle. Proof.
Table 3 shows that the Massey product h M h , h , h g i equals M h .Therefore, M h detects the Toda bracket h ηθ . , , σ θ i . The indeterminacy con-sists entirely of multiples of ηθ . . The Toda bracket contains θ h ηθ . , , σ i . Now h ηθ . , , σ i is zero because π , is zero.We have now shown that M h detects a multiple of η . In fact, it detects anon-zero multiple of η because M h cannot be hit by a differential by comparisonto the Adams spectral sequence for Cτ .Therefore, there exists a non-zero element of π , that is detected in Adamsfitration at most 12. The only possibility is that M survives. (cid:3) Lemma . d (∆ h h ) = τ h h d . Proof.
In the Adams E -page, ∆ h h equals h ∆ h , h , h i , with no indeter-minacy, because of the Adams differential d ( h ) = h h . Using that d (∆ h ) = τ h d , Moss’s higher Leibniz rule 2.27 implies that d (∆ h h ) is contained in h τ h d , h , h i + h ∆ h , , h i + h ∆ h , h , i . All of these brackets have no indeterminacy, and the last two equal zero. The firstbracket equals τ h h d , using the Adams differential d ( h ) = h h . (cid:3) Lemma . d ( P h d ) = 0 . Proof.
In the Adams E -page, the element P h d equals the Massey product h P d , h , h i , with no indeterminacy, because of the Adams differential d ( h ) = h h . Moss’s higher Leibniz rule 2.27 implies that d ( P h d ) is a linear combi-nation of multiples of h and of P d . The only possibility is that d ( P h d ) iszero. (cid:3) Lemma . d ( τ M P h d j ) = P ∆ h d . Proof.
Table 7 shows that d ( τ P d e ) = P d . Multiplication by τ M P h shows that d ( τ M P h d e ) equals τ M P h d . But τ M P h d e equals h · τ M P h d j , while τ M P h d is not divisible by h . Therefore, τ M P h d j cannotsurvive to the E -page.The possible values for d ( τ M P h d j ) are linear combinations of P ∆ h d and τ P ∆ h d e . Comparison to the Adams spectral sequence for tmf shows thatthe term τ P ∆ h d e cannot appear. (cid:3) Lemma . The element P h c is a permanent cycle. Proof.
Table 17 shows that P c detects the product ηρ . Using the MossConvergence Theorem 2.16 and the Adams differential d ( h ) = h h , the element P h c must survive to detect the Toda bracket h ηρ , , θ i . (cid:3) Remark . We suspect that P h c detects the product η ρ . However,the argument of Lemma 7.151 cannot be completed because the Toda bracket h ηρ , , θ i might have indeterminacy in lower Adams filtration. d differential Theorem . Table 7 lists some values of the Adams d differential on mul-tiplicative generators. The Adams d differential is zero on all multiplicative gen-erators not listed in the table. The list is complete through the 95-stem, exceptthat: (1) d (∆ M h ) might equal M P ∆ h e . Proof.
The d differential on many multiplicative generators is zero. A fewof these multiplicative generators appear in Table 7 because their proofs requirefurther explanation. For the remaining majority of such multiplicative generators,the d differential is zero because there are no possible non-zero values, or becauseof comparison to the Adams spectral sequences for Cτ , tmf , or mmf . In a few cases,the multiplicative generator is already known to be a permanent cycle as shown inTable 5. These cases do not appear in Table 7. .3. THE ADAMS d DIFFERENTIAL 41
The last column of Table 7 gives information on the proof of each differential.Most follow immediately by comparison to the Adams spectral sequence for Cτ , orby comparison to the classical Adams spectral sequence for tmf , or by comparisonto the C -motivic Adams spectral sequence for mmf .If an element is listed in the last column of Table 7, then the correspondingdifferential can be deduced from a straightforward argument using a multiplicativerelation. For example, d ( d · τ e g ) = d ( e · τ e g ) = e · P d = d , so d ( τ e g ) must equal d .The remaining more difficult computations are carried out in the followinglemmas. (cid:3) Table 7 lists the multiplicative generators of the Adams E -page through the95-stem whose d differentials are non-zero, or whose d differentials are zero fornon-obvious reasons. Lemma . d ( τ h · ∆ x ) = τ ∆ h d e . For completeness, we repeat the argument from [ , Remark 11.2]. Proof.
Table 7 shows that τ ∆ h g supports a d differential, and Table5 shows that τ ∆ h g + τ ∆ h g is a permanent cycle. Therefore, τ ∆ h g alsosupports a d differential.On the other hand, we have h · τ ∆ h g = P h · ∆ x = ∆ x h h , h h , h i . This expression equals h h · ∆ x, h h , h i by inspection of indeterminacies. There-fore, the Toda bracket h{ τ h · ∆ x } , σ, i cannot be well-formed, since otherwiseit would be detected by τ ∆ h g . The only possibility is that τ h · ∆ x is nota permanent cycle, and the only possible differential is that d ( τ h · ∆ x ) equals τ ∆ h d e . (cid:3) Lemma . d (∆ h ) = 0 . Proof.
Table 8 shows that d ( τ h · ∆ x ) equals τ d e . The element τ d e is not divisible by h in the E -page, so τ h · ∆ x cannot be divisible by h in the E -page.If d (∆ h ) equaled τ ∆ h d e , then ∆ h + τ h · ∆ x would survive to the E -page, and τ h · ∆ x would be divisible by h in the E -page. (cid:3) Lemma . d ( τ X ) = τ M h d . Proof.
Table 7 shows that d ( C ′ ) equals M h d . Therefore, either τ X or τ X + τ C ′ is non-zero on the E ∞ -page. The inclusion of the bottom cell into Cτ takes this element to h d e .In the homotopy of Cτ , there is a ν extension from h d e to τ B , and inclusionof the bottom cell into Cτ takes τ h C ′ to τ B .It follows that there must be a ν extension with target τ h C ′ . The only possi-bility is that τ X + τ C ′ is non-zero on the E ∞ -page, and therefore d ( τ X ) equals d ( τ C ′ ). (cid:3) Lemma . d ( h d ) = X . Proof.
The element X is a permanent cycle. The only possible target for adifferential is τ d e m , but this is ruled out by comparison to tmf .The element X must map to zero under the inclusion of the bottom cell into Cτ . Therefore, X is the target of a hidden τ extension, or it is hit by a differential.The only possible hidden τ extension would have source h · ∆ h . In Cτ ,there is an η extension from h d to h · ∆ h . Since h ∆ h maps non-trivially(to h · ∆ h ) under projection to the top cell of Cτ , it follows that h d also mapsnon-trivially under projection. The only possibility is that h d maps to h · ∆ h ,and therefore h · ∆ h does not support a hidden τ extension.Therefore, X must be hit by a differential, and there is just one possibility. (cid:3) Lemma . d ( M h g ) = 0 . Proof.
Table 3 shows that the Massey product h h g, h , g i equals M h g .The Moss Convergence Theorem 2.16 shows that M h g must survive to detect theToda bracket h{ h g } , , κ i . (cid:3) Lemma . d ( h G ) = τ g n . Proof.
Table 15 shows that there is a hidden 2 extension from h h g to τ gn .Therefore, τ gn detects 4 σκ .Table 3 shows that h h h , h , τ gn i consists of the two elements τ g n and τ g n + M h g · h h . Then the Toda bracket h η η , η, σκ i is detected by either τ g n or τ g n + M h g · h h . But M h g · h h is hit by an Adams d differential, so τ g n detects the Toda bracket.The Toda bracket has no indeterminacy, so it equals h η η , η, i σκ . This lastexpression must be zero.We have shown that τ g n must be hit by some differential. The only possibilityis that d ( h G ) = τ g n . (cid:3) Lemma . d (∆ h h g ) = τ M h d . Proof.
Table 8 shows that d ( A ′ ) = τ M h d . Now d A ′ is zero in the E -page, so τ M h d must also be zero in the E -page. (cid:3) Lemma . d (∆ h h g ) = τ ∆ h d e . Proof.
Table 11 shows that the element ∆ h d e detects the Toda bracket h τ ηκκ , η, η η i . Now shuffle to obtain τ h τ ηκκ , η, η η i = h τ, τ ηκκ , η i η η . Table 11 shows that h τ, τ ηκκ , η i is detected by h h h i . It follows that the expres-sion h τ, τ ηκκ , η i η η is zero, so τ ∆ h d e must be hit by some differential. Theonly possibility is that d (∆ h h g ) equals τ ∆ h d e . (cid:3) Lemma . d ( h e ) = τ h x , . Proof.
Table 21 shows that σ θ is detected by h h A . Since νσ = 0, theelement h h h A = τ h x , must be hit by a differential. The only possibility isthat d ( h e ) equals τ h x , . (cid:3) Lemma . d (∆ h h ) = τ d e l . .3. THE ADAMS d DIFFERENTIAL 43
Proof.
Table 17 shows that there is a hidden η extension from τ ∆ h g to τ d e m . Therefore, there is also a hidden η extension from τ ∆ h e g to τ d e l .Also, τ ∆ h e g detects an element in π , that is annihilated by τ . There-fore, τ d e l must be hit by some differential. Moreover, comparison to mmf showsthat τ d e l is not hit by a differential.The hidden η extension from τ ∆ h e g to τ d e l is detected by projectionfrom Cτ to the top cell. The only possibility is that this hidden η extension is theimage of the h extension from ∆ h h to ∆ h h in the Adams E ∞ -page for Cτ .Therefore, ∆ h h maps non-trivially under projection from Cτ to the top cell.Consequently, ∆ h h cannot be a permanent cycle in the Adams spectral sequencefor the sphere. (cid:3) Lemma . d (∆ j ) = τ M h e g . Proof.
Otherwise, both ∆ j and τ gC ′ would survive to the E ∞ -page, andneither could be the target of a hidden τ extension. They would both map non-trivially under inclusion of the bottom cell into Cτ . But there are not enoughelements in π , Cτ for this to occur. (cid:3) Lemma . The element τ h f is a permanent cycle. Proof.
Let α be an element of π , that is detected by τ h C ′ . Then να isdetected by τ h C ′ , and τ να is zero.Let α be an element of π , Cτ that is detected by h h h . Projection from Cτ to the top cell takes α to να . Moreover, in the homotopy of Cτ , the Todabracket h , σ , α i is detected by h c .Now projection from Cτ to the top cell takes h , σ , α i to h , σ , να i , whichequals zero by Lemma 6.23. Therefore, h c maps to zero under projection to thetop cell of Cτ , so it must be in the image of inclusion of the bottom cell. The onlypossibility is that τ h f survives and maps to h c under inclusion of the bottomcell. (cid:3) Remark . In the proof of Lemma 5.46, we have used that d ( τ p + h h h )equals τ h C ′ in order to conclude that τ να is zero. This differential depends onwork in preparation [ ].However, we can also prove Lemma 5.46 independently of [ ]. Lemma 5.57shows that the other possible value of d ( τ p + h h h ) is τ h C ′ + τ h (∆ e + C ).In this case, let β be an element of π , that is detected by ∆ e + C . Then να + σβ is detected by τ h C ′ + h (∆ e + C ), and τ ( να + σβ ) is zero.Projection from Cτ to the top cell takes α to να + σβ , and takes h , σ , α i to h , σ , να + σβ i , which equals zero by Lemmas 6.23 and 6.24. As in the proof ofLemma 5.46, the only possibility is that τ h f survives and maps to h c underinclusion of the bottom cell of Cτ . Lemma . d ( h c ) = τ h h h Q . Proof.
Lemma 7.155 shows that there exists an element α in π , that isdetected by h Q + h n such that τ να equals ( ησ + ǫ ) θ .Table 11 shows that the Toda bracket h ν, σ, σ i is detected by h h , so theelement τ h h h Q detects τ α h ν, σ, σ i , which is contained in h τ να, σ, σ i . Theindeterminacy in these expressions is zero because τ να · π , and 2 σ · π , areboth zero. We now know that τ h h h Q detects the Toda bracket h ( ǫ + ησ ) θ , σ, σ i . Thisbracket contains θ h ǫ + ησ, σ, σ i . Lemma 6.6 shows that the bracket h ǫ + ησ, σ, σ i contains 0, so θ h ǫ + ησ, σ, σ i equals zero.Finally, we have shown that τ h h h Q detects zero, so it must be hit by somedifferential. (cid:3) Lemma . d ( x , ) = 0 . Proof.
Consider the exact sequence π , → π , Cτ → π , . The middle term π , Cτ is isomorphic to ( Z / . The elements of π , that arenot divisible by τ are detected by P h c , and possibly x , and τ ∆ h H . Onthe other hand, the elements of π , that are annihilated by τ are detected by τ ∆ c e g and possibly M ∆ h e .In order for the possibility M ∆ h e to occur, either x , or τ ∆ h H wouldhave to support a differential hitting τ M ∆ h e , in which case one of those possi-bilities could not occur.If d ( x , ) equaled τ gG , then there would not be enough elements to makethe above sequence exact. (cid:3) Lemma . d ( τ ∆ h H ) = 0 . Proof.
The element ∆ h d is a permanent cycle that cannot be hit by anydifferential because h · ∆ h d cannot be hit by a differential. The element ∆ h d cannot be in the image of projection from Cτ to the top cell, and it cannot supporta hidden τ extension. Therefore, τ ∆ h d cannot be hit by a differential. (cid:3) Lemma . d ( τ h B g ) = M h d . Proof.
Table 21 shows that
M d detects κθ . . Therefore, M d detects κ θ . ,which equals η κ θ . because Table 17 shows that there is a hidden η extensionfrom τ h g to d .Now η κ θ . is zero because η κθ . is zero. Therefore, M d and M h d mustboth be hit by differentials.There are several possible differentials that can hit M h d . The element h x , cannot be the source of this differential because Table 5 shows that x , is a permanent cycle. The element τ h gC ′ cannot be the source of the differentialbecause h gC ′ is a permanent cycle by comparison to mmf . The element ∆ h g g cannot be the source because it equals h (∆ e + C ) g . The only remaining possi-bility is that d ( τ h B g ) equals M h d . (cid:3) Lemma . d (∆ h A ′ ) = 0 . Proof.
In the Adams E -page, the element ∆ h A ′ equals the Massey product h A ′ , h , τ d i , with no indeterminacy because of the Adams differential d (∆ h ) = τ h d . Moss’s higher Leibniz rule 2.27 implies that d (∆ h A ′ ) is contained in h , h , τ d i + h A ′ , , τ d i + h A ′ , h , i , so it is a linear combination of multiples of A ′ and τ d . The only possibility is that d (∆ h A ′ ) is zero. (cid:3) Lemma . d ( h h ) = h g . .4. THE ADAMS d DIFFERENTIAL 45
Proof.
By comparison to the Adams spectral sequence for Cτ , the value of d ( h h ) is either h g or h g + τ h h D .Table 21 shows that h g detects the product θ θ . Since 2 θ θ equals zero, h g must be hit by a differential. (cid:3) d differential Theorem . Table 8 lists some values of the Adams d differential on mul-tiplicative generators. The Adams d differential is zero on all multiplicative gen-erators not listed in the table. The list is complete through the 95-stem, exceptthat: (1) d (∆ g ) might equal τ ∆ h g . Proof.
The d differential on many multiplicative generators is zero. For themajority of such multiplicative generators, the d differential is zero because thereare no possible non-zero values, or by comparison to the Adams spectral sequencefor Cτ , or by comparison to tmf or mmf . In a few cases, the multiplicative gener-ator is already known to be a permanent cycle; h h is one such example. A fewadditional cases appear in Table 8 because their proofs require further explanation.The last column of Table 8 gives information on the proof of each differen-tial. Many computations follow immediately by comparison to the Adams spectralsequence for Cτ .If an element is listed in the last column of Table 8, then the correspondingdifferential can be deduced from a straightforward argument using a multiplicativerelation. For example, d ( τ · gA ′ ) = d ( τ g · A ′ ) = τ g · τ M h d = τ M h e , so d ( gA ′ ) must equal τ M h e .A few of the more difficult computations appear in [ ]. The remaining moredifficult computations are carried out in the following lemmas. (cid:3) Table 8 lists the multiplicative generators of the Adams E -page through the95-stem whose d differentials are non-zero, or whose d differentials are zero fornon-obvious reasons. Lemma . d ( τ h · ∆ x ) = τ d e . Proof.
The element τ d e cannot be hit by a differential. There is a hidden η extension from τ ∆ h d e to τ d e because of the hidden τ extensions from τ h g + h h c e to ∆ h d e and from h h c e to d e . This shows that τ d e must be hit by some differential.This hidden η extension is detected by projection from Cτ to the top cell. Since P h c d in Cτ maps to τ ∆ h d e under projection to the top cell, it follows that P h h c d in Cτ maps to τ d e under projection to the top cell.If τ h · ∆ x survived, then it could not be the target of a hidden τ extensionand it could not be hit by a differential. Also, it could not map non-trivially underinclusion of the bottom cell into Cτ , since the only possible value P h h c d hasalready been accounted for in the previous paragraph. (cid:3) Lemma . d ( h d i ) = τ ∆ h d . Proof.
We showed in Lemma 5.14 that
P h h j cannot be divisible by h inthe E ∞ -page. Therefore, h d i must support a differential. (cid:3) Lemma . d ( τ p + h h h ) equals either τ h C ′ or τ h C ′ + τ h (∆ e + C ) . Proof.
Projection to the top cell of Cτ takes h D to τ d g . Moreover, thereis a ν extension in the homotopy of Cτ from h h h to h D . Therefore, this ν extension must be in the image of projection to the top cell.Table 19 shows that there is a hidden ν extension from τ h C ′ to τ d g . There-fore, either τ h C ′ or τ h C ′ + h (∆ e + C ) is in the image of projection to the topcell, so τ h C ′ or τ h C ′ + τ h (∆ e + C ) is hit by a differential. The element τ p + h h h is the only possible source for this differential. (cid:3) Lemma . d ( h x , ) = 0 . Proof.
Table 14 shows that there is a hidden τ extension from M h h g to M h d . Therefore, M h g must also support a τ extension. This shows that τ M h g cannot be the target of a differential. (cid:3) Lemma . d ( h D ) = τ d g . Proof.
Suppose for sake of contradiction that h D survived, and let α be anelement of π , that is detected by it. Table 14 shows that there is a hidden τ extension from h h c to h h D . Therefore, h h D detects both 2 α and τ ηǫη .However, it is possible that the difference between these two elements is detectedby τ M d or by τ ∆ h d e . We will handle of each of these cases.First, suppose that 2 α equals τ ηǫη . Then the Toda bracket (cid:28) η, (cid:2) τ ηǫ (cid:3) , (cid:20) αη (cid:21)(cid:29) is well-defined. Inclusion of the bottom cell into Cτ takes this bracket to (cid:28) η, (cid:2) (cid:3) , (cid:20) αη (cid:21)(cid:29) = h η, , α i , so h η, , α i is in the image of inclusion of the bottom cell.On the other hand, in the homotopy of Cτ , the bracket h η, , α i is detectedby h h c , with indeterminacy generated by h h Q . These elements map non-trivially under projection to the top cell, which contradicts that they are in theimage of inclusion of the bottom cell.Next, suppose that 2 α + τ ηǫη is detected by τ ∆ h d e . Then the Todabracket * η, (cid:2) τ ηǫ τ β (cid:3) , αη κ + is well-defined, where β is an element of π , that is detected by ∆ h d . Thesame argument involving inclusion of the bottom cell into Cτ applies to this Todabracket.Finally, assume that 2 α + τ ηǫη is detected by τ M d . Table 21 shows that M d detects κθ . , so τ M d detects τ κ θ . . Then 2 α + τ ηǫη equals either τ κ θ . or τ κ θ . + τ βκ . We can apply the same argument to the Toda bracket * η, (cid:2) τ ηǫ τ κθ . (cid:3) , αη κ + , .4. THE ADAMS d DIFFERENTIAL 47 or to the Toda bracket * η, (cid:2) τ ηǫ τ κθ . τ β (cid:3) , αη κκ + . We have now shown by contradiction that h D does not survive. After rul-ing out other possibilities by comparison to Cτ and to mmf , the only remainingpossibility is that d ( h D ) equals τ d g . (cid:3) Lemma . d ( τ gG ) = τ M ∆ h d . Proof.
Suppose for sake of contradiction that the element τ gG survived. Itcannot be the target of a hidden τ extension, and it cannot be hit by a differential.Therefore, it maps non-trivially under inclusion of the bottom cell into Cτ , and theonly possible image is ∆ e + τ ∆ h e g .Let α be an element of π , that is detected by τ gG . Consider the Todabracket h α, ν, ν i . Lemma 4.12 implies that this Toda bracket is detected by e x , ,or is detected in higher Adams filtration.On the other hand, under inclusion of the bottom cell into Cτ , the Toda bracketis detected by ∆ h g . This is inconsistent with the conclusion of the previousparagraph, since inclusion of the bottom cell can only increase Adams filtrations.We now know that τ gG does not survive. After eliminating other possibilitiesby comparison to mmf , the only remaining possibility is that d ( τ gG ) equals τ M ∆ h d . (cid:3) Lemma . d ( g ) = h d . Proof.
Table 11 shows that h h detects the Toda bracket h η, , θ i . There-fore, h h d detects κ h η, , θ i . Now consider the shuffle τ κ h η, , θ i = h τ κ , η, i θ . Lemma 6.7 shows that the last bracket is zero. Therefore, h h d does not supporta hidden τ extension, so it is either hit by a differential or in the image of projectionfrom Cτ to the top cell.In the Adams spectral sequence for Cτ , the element h h h detects the Todabracket h θ , , θ i . Therefore, h h h must be in the image of inclusion of thebottom cell into Cτ . In particular, h h h cannot map to h h d under projectionfrom Cτ to the top cell.Now h h d cannot be in the image of projection from Cτ to the top cell, soit must be hit by some differential. The only possibility is that d ( h g ) equals h h d . (cid:3) Lemma . d ( e x , ) = M ∆ h c d . Proof. If M ∆ h c d were a permanent non-zero cycle, then it could not sup-port a hidden τ extension because Lemma 5.80 shows that M P ∆ h d is hit bysome differential. Therefore, it would lie in the image of projection from Cτ to thetop cell, and the only possible pre-image is the element ∆ h g in the E ∞ -page ofthe Adams spectral sequence for Cτ .There is a σ extension from ∆ e + τ ∆ h e g to ∆ h g in the Adams spectralsequence for Cτ . Then M ∆ h c d would also have to be the target of a σ extension.The only possible source for this extension would be M ∆ h d . Table 17 shows that
M h detects ηθ . , so M ∆ h d detects ηθ . { ∆ h d } . Theproduct ησθ . { ∆ h d } equals zero because σ { ∆ h d } is zero. Therefore, M ∆ h d cannot support a hidden σ extension to M ∆ h c d .We have now shown that M ∆ h c d must be hit by some differential, and theonly possibility is that equals d ( e x , ). (cid:3) Theorem . Table 9 lists some values of the Adams d r differential on multi-plicative generators of the E r -page, for r ≥ . For r ≥ , the Adams d r differentialis zero on all multiplicative generators of the E r -page not listed in the table. Thelist is complete through the 90-stem, except that:(1) d ( τ h g + τ h e ) might equal ∆ h n .(2) d ( x , ) might equal M ∆ h d .(3) d ( h x , ) might equal M ∆ h d .(4) d ( h h g ) or d ( h h g + h f ) might equal M ∆ h d .(5) d ( τ ∆ h H ) might equal τ M ∆ h e .(6) d ( x , ) might equal τ M ∆ h e .(7) d (∆ f ) might equal τ M d . Proof.
The d r differential on many multiplicative generators is zero. For themajority of such multiplicative generators, the d r differential is zero because thereare no possible non-zero values, or by comparison to the Adams spectral sequencefor Cτ , or by comparison to tmf or mmf . In a few cases, the multiplicative generatoris already known to be a permanent cycle, as shown in Table 5. A few additionalcases appear in Table 9 because their proofs require further explanation.Some of the more difficult computations appear in [ ]. The remaining moredifficult computations are carried out in the following lemmas. (cid:3) Table 9 lists the multiplicative generators of the Adams E r -page, for r ≥ d r differentials are non-zero, or whose d r differentialsare zero for non-obvious reasons. Remark . Because d (∆ g g ) equals M d , the uncertainty about d (∆ f )is inconsequential. Either ∆ f or ∆ f + τ ∆ g g survives to the E ∞ -page. Lemma . (1) d ( τ Q + τ n ) = 0 .(2) d ( gQ ) = 0 Proof.
Several possible differentials on these elements are eliminated by com-parison to the Adams spectral sequences for Cτ and for tmf . The only remainingpossibility is that d ( τ Q + τ n ) might equal τ M h g , and that d ( gQ ) mightequal τ M h g .The element M ∆ h e is not hit by any differential because Table 5 shows that h c is a permanent cycle, and Table 11 shows that τ gQ = h Q must survive todetect the Toda bracket h θ , τ κ, { t }i .Lemma 6.25 shows that M ∆ h e detects the Toda bracket h τ ηκ , , κ i , whichcontains τ κ h η, , κ i . Lemma 6.10 shows that this expression contains zero. Wenow know that M ∆ h e detects an element in the indeterminacy of the bracket h τ ηκ , , κ i . In fact, it must detect a multiple of τ ηκ since 2 κ · π , is zero. .5. HIGHER DIFFERENTIALS 49 The only possibility is that M ∆ h e detects κ times an element detected by τ M h g . Therefore, τ M h g cannot be hit by a differential. This shows that τ Q + τ n is a permanent cycle.We also know that M ∆ h e is the target of a hidden τ extension, since itdetects a multiple of τ . The element τ M h g is the only possible source of thishidden τ extension, so it cannot be hit by a differential. This shows that d ( gQ )cannot equal τ M h g . (cid:3) Lemma . (1) d ( h H ) = M c d .(2) d ( τ h H ) = M P d . Proof.
Table 3 shows that
M P d equals the Massey product h P d , h , g i .This implies that M P d detects the Toda bracket h τ η κ, , κ i . Lemma 6.16 showsthat this Toda bracket consists entirely of multiples of τ η κ .We now know that M P d detects a multiple of τ η κ . The only possibility isthat M P d detects η times an element detected by τ M h g .We will show in Lemma 7.108 that τ M h g is the target of a ν extension, so τ M h g cannot support a hidden η extension. Therefore, M P d must be hit bysome differential. The only possibility is that d ( τ h H ) equals M P d . Then h H cannot survive to the E -page, so d ( h H ) equals M c d . (cid:3) Lemma . The element τ h p is a permanent cycle. Proof.
Lemma 5.57, together with results of [ ], show that τ h p survives tothe E -page. We must eliminate possible higher differentials.Table 14 shows that there is a hidden τ extension from τ h C ′′ to ∆ h h c .This means that τ h C ′′ + h h (∆ e + C ) must also support a hidden τ extension.The two possible targets for this hidden τ extension are ∆ h c and τ ∆ h g + τ ∆ h g . The second possibility is ruled out by comparison to tmf , so ∆ h c cannot be hit by a differential. (cid:3) Lemma . The element
P h h h is a permanent cycle. Proof.
First note that projection from Cτ to the top cell takes P h h to anon-zero element. If P h h h were not a permanent cycle in the Adams spectralsequence for the sphere, then projection from Cτ to the top cell would also take P h h h to a non-zero element. Then the 2 extension from P h h to P h h h in π , Cτ would project to a 2 extension in π , . However, there are no possible 2extensions in π , . (cid:3) Lemma . d ( m ) = 0 . Proof.
The only other possibility is that d ( m ) equals τ g t . If that werethe case, then the ν extension from τ g t to τ c g would be detected by projectionfrom Cτ to the top cell. However, the homotopy groups of Cτ have no such ν extension. (cid:3) Lemma . d ( h x ) = 0 . Proof.
Table 5 shows that τ h x is a permanent cycle. Then d ( τ h x ) can-not equal τ M e , and d ( h x ) cannot equal τ M e . (cid:3) Lemma . d ( h h h ) = 0 . Proof.
Table 5 shows that h h h is a permanent cycle. Therefore, the Adamsdifferential d ( h h h ) does not equal τ h c A ′ , and d ( h h h ) does not equal τ c A ′ . (cid:3) Lemma . d ( x , ) = 0 . Proof. If τ ∆ h d were hit by a differential, then the ν extension from ∆ h d to ∆ h h d would be detected by projection from Cτ to the top cell. But thehomotopy of Cτ has no such ν extension. (cid:3) Lemma . The element x , is a permanent cycle. Proof.
In the Adams spectral sequence for Cτ , there is a hidden η extensionfrom h x , to x , . Therefore, x , lies in the image of inclusion of the bottomcell into Cτ . The only possible pre-image is the element x , in the Adams spectralsequence in the sphere, so x , must survive. (cid:3) Lemma . (1) d ( h gH ) = M c e .(2) d ( τ h gH ) = 0 . Proof. If M c e is non-zero in the E ∞ -page, then it detects an element that isannihilated by τ because Lemma 5.75 shows that the only possible target of such anextension is hit by a differential. Then M c e would be in the image of projectionfrom Cτ to the top cell. The only possible pre-image would be the element ∆ g g of the Adams spectral sequence for Cτ .In the Adams spectral sequence for Cτ , there is a σ extension from gA ′ to∆ g g . Projection from Cτ to the top cell would imply that there is a hidden σ extension in the homotopy groups of the sphere, from M h e to M c e , because gA ′ maps to M h e under projection from Cτ to the top cell.But M h e detects ηθ . { e } , which cannot support a σ extension. This estab-lishes the first formula.For the second formula, if d ( τ h gH ) were equal to τ ∆ h e g , then the sameargument would apply, with τ ∆ h e g substituted for M c e . (cid:3) Lemma . d (∆ g g ) = M d . Proof.
The proof of Lemma 5.51 shows that
M d must be hit by a differential.The only possibility is that d (∆ g g ) equals M d .Alternatively, Lemma 5.66 shows that d ( τ h H ) = M P d . Note that τ g · τ h H = 0 in the E -page. Therefore, τ M d = τ g · M P d must already be zero inthe E -page. The only possibility is that d ( τ ∆ g g ) = τ M d , and then d (∆ g g ) = M d . (cid:3) Lemma . The element h g is a permanent cycle. Proof.
In the homotopy of Cτ , the product θ θ is detected by h g . In thesphere, the product θ θ is therefore non-zero and detected in Adams filtration atmost 6.Table 11 shows that the Toda bracket h , θ , θ , i contains θ . Therefore, theproduct θ θ is contained in θ h , θ , θ , i = h θ , , θ , θ i . .5. HIGHER DIFFERENTIALS 51 (Note that the sub-bracket h θ , θ , i is zero because π , is zero.) Therefore, θ θ is divisible by 2. It follows that θ θ is detected by h g , and h g is a permanentcycle that detects h θ , , θ , θ i . (cid:3) Lemma . d (∆ h e ) = 0 . Proof.
Consider the element τ M h g in the Adams spectral sequence for Cτ . This element cannot be in the image of inclusion of the bottom cell into Cτ .Therefore, it must map non-trivially under projection from Cτ to the top cell. Theonly possibility is that τ M h g is the image. Therefore, τ M h g cannot be thetarget of a differential. (cid:3) Lemma . d ( x , ) does not equal τ ∆ h g . Proof. If τ ∆ h g were hit by a differential, then the 2 extension from τ ∆ h g to τ ∆ h h g would be detected by projection from Cτ to the top cell.But the homotopy of Cτ has no such 2 extension. (cid:3) Lemma . d (∆ h e ) = 0 . Proof.
Consider the element e x , in the Adams E ∞ -page for Cτ . It cannotbe in the image of inclusion of the bottom cell into Cτ , so it must project to a non-zero element in the top cell. The only possible image is M ∆ h g . Therefore, M ∆ h g cannot be the target of a differential. (cid:3) Lemma . The element
M P ∆ h d is hit by some differential. Proof.
Table 14 shows that there is a hidden τ extension from ∆ h c d to P ∆ h d . Therefore, P ∆ h d detects τ ǫ { ∆ h d } . On the other hand, Tables 17and 21 show that P ∆ h d also detects τ ηκ { ∆ h h } . Since there are no elementsin higher Adams filtration, we have that τ ǫ { ∆ h d } equals τ ηκ { ∆ h h } .Table 21 shows that M P detects τ ǫθ . , so M P ∆ h d detects τ ǫ { ∆ h d } θ . ,which equals τ ηκ { ∆ h h } θ . . But τ ηκθ . is zero because all elements of π , are detected by tmf . This shows that M P ∆ h d detects zero, so it must be hit bya differential. (cid:3) HAPTER 6
Toda brackets
The purpose of this chapter is to establish various Toda brackets that are usedelsewhere in this manuscript. Many Toda brackets can be easily computed fromthe Moss Convergence Theorem 2.16. These are summarized in Table 11 withoutfurther discussion. However, some brackets require more complicated arguments.Those arguments are collected in this chapter.We will need the following C -motivic version of a theorem of Toda [ , Theorem3.6] that applies to symmetric Toda brackets. Theorem . Let α be an element of π s,w , with s even. There exists anelement α ∗ in π s +1 , w such that h α, β, α i contains the product βα ∗ for all β suchthat αβ . Corollary . If β = 0 , then h , β, i contains τ ηβ . Proof.
Apply Theorem 6.1 to α = 2. We need to find the value of α ∗ . Table3 shows that the Massey product h h , h , h i equals τ h . The Moss ConvergenceTheorem 2.16 then shows that h , η, i equals τ η . It follows that α ∗ equals τ η . (cid:3) Theorem . Table 11 lists some Toda brackets in the C -motivic stable ho-motopy groups. Proof.
The fourth column of the table gives information about the proof ofeach Toda bracket.If the fourth column shows a Massey product, then the Toda bracket followsfrom the Moss Convergence Theorem 2.16. If the fourth column shows an Adamsdifferential, then the Toda bracket follows from the Moss Convergence Theorem2.16, using the mentioned differential.A few Toda brackets are established elsewhere in the literature; specific citationsare given in these cases.Additional more difficult cases are established in the following lemmas. (cid:3)
Table 11 lists information about some Toda brackets. The third column ofTable 11 gives an element of the Adams E ∞ -page that detects an element of theToda bracket. The fourth column of Table 11 gives partial information aboutindeterminacies, again by giving detecting elements of the Adams E ∞ -page. Wehave not completely analyzed the indeterminacies of all brackets when the detailsare inconsequential for our purposes. The fifth column indicates the proof of theToda bracket, and the sixth column shows where each specific Toda bracket is usedin the manuscript. Lemma . The Toda bracket h κ, , η i contains zero, with indeterminacy gen-erated by ηρ .
534 6. TODA BRACKETS
Proof.
Using the Adams differential d ( h h ) = h d , the Moss ConvergenceTheorem 2.16 shows that the Toda bracket is detected in filtration at least 3. Theonly element in sufficiently high filtration is P c , which detects the product ηρ .This product lies in the indeterminacy, so the bracket must contain zero. (cid:3) Lemma . The Toda bracket h κ, , η, ν i is detected by τ g . Proof.
The subbracket h , η, ν i is strictly zero, since π , is zero. The sub-bracket h κ, , η i contains zero by Lemma 6.4. Therefore, the fourfold bracket h κ, , η, ν i is well-defined.Shuffle to obtain h κ, , η, ν i η = κ h , η, ν, η i . Table 11 shows that ǫ is contained in the Toda bracket h η , ν, η, i , so the latterexpression equals ǫκ , which is detected by c d . It follows that h κ, , η, ν i must bedetected by τ g . (cid:3) Lemma . The Toda bracket h ǫ + ησ, σ, σ i contains zero, with indeterminacygenerated by νκ in { P h d } . Proof.
Consider the shuffle h ǫ + ησ, σ, σ i η = ( ǫ + ησ ) h σ, σ, η i . Table 11 shows that h h detects h σ, σ, η i , so h h c detects the product ǫ h σ, σ, η i .On the other hand, Table 21 shows that h h c also detects ησ h σ, σ, η i . Therefore,( ǫ + ησ ) h σ, σ, η i is detected in filtration greater than 5. Then h c cannot detect h ǫ + ησ, σ, σ i .The shuffle 2 h ǫ + ησ, σ, σ i = h , ǫ + ησ, σ i σ = 0shows that no elements of the Toda bracket can be detected by τ h g or τ h h g .The element 4 νκ generates the indeterminacy because it equals τ ηκ ( ǫ + ησ ). (cid:3) Lemma . The Toda bracket h τ κ , η, i equals zero, with no indeterminacy. Proof.
The Adams differential d (∆ h ) = τ h d implies that the bracket isdetected by h · ∆ h , which equals zero. Therefore, the Toda bracket is detectedin Adams filtration at least 7, but there are no elements in the Adams E ∞ -page insufficiently high filtration.The indeterminacy can be computed by inspection. (cid:3) Lemma . The Toda bracket h η , θ , η i contains zero, with indeterminacygenerated by η η . Proof.
If the bracket were detected by h d , then ν h η , θ , η i = h ν, η , θ i η would be detected by h d . However, h d does not detect a multiple of η .The bracket cannot be detected by τ h e by comparison to tmf .By inspection, the only remaining possibility is that the bracket contains zero.The indeterminacy can be computed by inspection. (cid:3) Lemma . The Toda bracket h τ, η κ , η i is detected by t , with indeterminacygenerated by η µ . . TODA BRACKETS 55 Proof.
There is a relation h · h d = t in the homotopy of Cτ . Using theconnection between Toda brackets and cofibers as described in [ , Section 3.1.1],this shows that t detects the Toda bracket.The indeterminacy is computed by inspection. (cid:3) Lemma . The Toda bracket h η, , κ i contains zero. Proof.
The Massey product
M h = h h , h , h g i shows that M h detects theToda bracket. Table 17 shows that M h , ∆ h c , and τ d l + ∆ c d are all targetsof hidden η extensions. (Beware that the hidden η extension from h h to M h is acrossing extension in the sense of Section 2.1, but that does not matter.) Therefore, M h detects only multiples of η , so the Toda bracket contains a multiple of η . Thisimplies that it contains zero, since multiples of η belong to the indeterminacy. (cid:3) Lemma . The Toda bracket h τ κ , σ , i equals zero. Proof.
No elements of the bracket can be detected by τ ∆ h d g by compar-ison to tmf .Consider the shuffle h τ κ , σ , i κ = τ κ h σ , , κ i . The bracket h σ , , κ i is zero because it is contained in π , = 0. On the otherhand, while { τ M d } κ is non-zero and detected by τ M d . Therefore, no elementsof h τ κ , σ , i can be can be detected by τ M d . (cid:3) Lemma . For every α that is detected by h h , the Toda bracket h , σ , α i contains zero. The indeterminacy is generated by τ κ , which is detected by τ d l . Proof.
The product σ α is zero for every α that is detected by h h . Fordegree reasons, the only elements that could detect this product either support η extensions or are detected by tmf . Therefore, the bracket is defined for all α .By comparison to tmf , the bracket cannot be detected by τ g . Table 15 showsthat τ d l is the target of a hidden 2 extension, so it detects an element in theindeterminacy. Since there are no other possibilities, the bracket must containzero. (cid:3) Remark . This result is consistent with Table 23 of [ ], which claims thatthe bracket h , σ , θ . i contains an element that is detected by B . The element B is now known to be zero in the Adams E ∞ -page, so this just means that thebracket contains an element detected in Adams filtration strictly greater than thefiltration of B . Lemma . The Toda bracket h θ , η , θ i equals zero. Proof.
Theorem 6.1 says that there exists an element θ ∗ in π , such that h θ , η , θ i contains η θ ∗ . The group π , is zero, so θ ∗ must be zero, and thebracket must contain zero.In order to compute the indeterminacy of h θ , η , θ i , we must consider theproduct of θ with elements of π , . There are several cases to consider.First consider { ∆ h h } . The product θ { ∆ h h } is detected in Adams filtra-tion at least 10, but there are no elements in sufficiently high filtration.Next consider νθ detected by p . The product θ is zero [ ], so νθ is alsozero. Finally, consider ηη detected by h h . Table 11 shows that h η, , θ i detects η . Shuffle to obtain ηη θ = η h η, , θ i θ = η h , θ , θ i . The bracket h , θ , θ i is zero because it is contained in π , = 0. (cid:3) Lemma . The Toda bracket h η , θ , η , θ i is detected by ∆ h . Proof.
Table 3 shows that ∆ h equals h h , h , h , h i . Therefore, ∆ h de-tects h η , θ , η , θ i , if the Toda bracket is well-defined.In order to show that the Toda bracket is well-defined, we need to know that thesubbrackets h η , θ , η i and h θ , η , θ i contain zero. These are handled by Lemmas6.8 and 6.14. (cid:3) Lemma . The Toda bracket h τ η κ, , κ i contains zero, and its indetermi-nacy is generated by multiples of τ η κ . Proof.
The bracket h τ η κ, , κ i contains τ ηκ h η, , κ i . Lemma 6.10 showsthat this expression contains zero.It remains to show that κ · π , equals zero. There are several cases toconsider.First, the product τ ση κ in π , could only be detected by τ g or τ d l .Comparison to tmf rules out both possibilities. Therefore, τ ση κ is zero.Second, the product κκ in π , must be detected in filtration at least 9, since τ gg equals zero, so it could only be detected by h (∆ e + C ). This implies that τ νκκ is zero.Third, we must consider the product ρ κ . Table 11 shows that the Todabracket h σ, , ρ i detects ρ . Then ρ κ is contained in h σ, , ρ i κ = σ h , ρ , κ i . The latter bracket is contained in π , . As above, comparison to tmf shows thatthe expression is zero. (cid:3) Lemma . The Toda bracket h η, ν, τ θ . κ i is detected by τ h D ′ . Proof.
Table 6 (see also Remark 5.10) shows that either d ( τ D ′ ) or d ( τ D ′ + τ h G ) equals τ M h g . In either case, the Moss Convergence Theorem 2.16 showsthat τ h D ′ detects the Toda bracket. (cid:3) Lemma . There exists an element α in π , detected by τ h C ′ such thatthe Toda bracket h η, ν, α i is defined and detected by τ h p . Proof.
The differential d ( τ p + h h h ) = τ h C ′ and the Moss ConvergenceTheorem 2.16 establish that the Toda bracket is detected by τ h p , provided thatthe Toda bracket is well-defined.Let α be an element of π , that is detected by τ h C ′ . Then να does notnecessarily equal zero; it could be detected in higher filtration by τ h B + h D ′ .Then we can adjust our choice of α by an element detected by τ B + D ′ to ensurethat να is zero. (cid:3) Lemma . The Toda bracket h σ , , { t } , τ κ i is detected by h Q + h D . . TODA BRACKETS 57 Proof.
The subbracket h σ , , { t }i contains zero by comparison to Cτ , andits indeterminacy is generated by σ θ = 4 σκ detected by τ gn . The subbracket h , { t } , τ κ i is strictly zero because it cannot be detected by h h h i by comparisonto tmf . This shows that the desired four-fold Toda bracket is well-defined.Consider the relation η h σ , , { t } , τ κ i ⊆ hh η, σ , i , { t } , τ κ i . Let α be any element of h η, σ , i . Table 11 shows that α is detected by h h and equals either η or η + ηρ . By inspection, the indeterminacy of h α, { t } , τ κ i equals τ κ · π , , which is detected in Adams filtration at least 14. (In fact, theindeterminacy is non-zero, since it contains both τ κ · { M c } detected by τ M d andalso τ κ · { ∆ h d } detected by τ ∆ h d e .)Table 11 shows that h α, { t } , τ κ i is detected by h h Q . Together with thepartial analysis of the indeterminacy in the previous paragraph, this shows that h α, { t } , τ κ i does not contain zero.Then η h σ , , { t } , τ κ i also does not contain zero, and the only possibility is that h σ , , { t } , τ κ i is detected by h Q + h D . (cid:3) Lemma . The Toda bracket h θ , θ , κ i equals zero. Proof.
The Massey product h h , h , d i equals zero, since h h h , h , d i = h h , h , h i d = 0 , while h x , is not zero. The Moss Convergence Theorem 2.16 then implies that h θ , θ , κ i is detected in Adams filtration at least 8.The only element in sufficiently high filtration is P h h . However, η h θ , θ , κ i = h η , θ , θ i κ = 0 , while h · P h h is not zero. Then h θ , θ , κ i must contain zero because there areno remaining possibilities.The indeterminacy can be computed by inspection, using that θ θ . is zero bycomparison to Cτ . (cid:3) Lemma . The Toda bracket h κ, , θ i is detected by h d . Proof.
The differential d ( h h ) = h d implies that h κ, , θ i is detected by h h · h = 0 in filtration 4. In other words, the Toda bracket is detected in Adamsfiltration at least 5.The element h h d detects h ηκ, , θ i , using the Adams differential d ( h ) = h h . This expression contains η h κ, , θ i , which shows that h κ, , θ i is detected infiltration at most 5.The only possibility is that the Toda bracket is detected by h d . (cid:3) Lemma . The Toda bracket h , η, τ η { h x , }i is detected by τ h x . Proof.
Let α be an element of π , that is detected by h x , . First wemust show that the Toda bracket is well-defined.Note that 2 α is zero because there are no 2 extensions in π , in sufficientlyhigh Adams filtration. Now consider the shuffle τ η α = h , η, i α = 2 h η, , α i . Table 11 shows that h η, , α i is detected by h h x , , but this element does notsupport a hidden 2 extension. This shows that τ η α is zero and that the Todabracket is well-defined.Finally, use the Adams differential d ( h e ) = τ h x , and the relation h · h e = τ h x to compute the Toda bracket. (cid:3) Lemma . The Toda bracket h , σ , { τ h C ′ }i equals zero, with no indeter-minacy. Proof.
Let α be an element of π , that is detected by τ h C ′ , so να is theunique element that is detected by τ h C ′ . We consider the Toda bracket h , σ , να i .By inspection, the indeterminacy is zero, so the bracket equals h , σ , ν i α , whichequals h α, , σ i ν .Apply the Moss Convergence Theorem 2.16 with the Adams d differential tosee that the Toda bracket h α, , σ i is detected by 0 in Adams filtration 9, but itcould be detected by a non-zero element in higher filtration. However, this showsthat h α, , σ i ν is zero by inspection. (cid:3) Lemma . The Toda bracket h , σ , { h (∆ e + C ) }i equals zero, with noindeterminacy. Proof.
Let β be an element of π , that is detected by ∆ e + C , so σβ is the unique element that is detected by h (∆ e + C ). We consider the Todabracket h , σ , σβ i . By inspection, the indeterminacy is zero, so the bracket equals h , σ , β i σ .Apply the Moss Convergence Theorem 2.16 with the Adams d differential tosee that the Toda bracket h , σ , β i is detected by 0 in Adams filtration 9, but itcould be detected by a non-zero element in higher filtration. Then the only possiblenon-zero value for h , σ , β i σ is { M ∆ h h } σ . Table 21 shows that M ∆ h h detects { ∆ h h } θ . , so σ { M ∆ h h } equals σ { ∆ h h } θ . , which equals zero. (cid:3) Lemma . The Toda bracket h τ ηκ , , κ i is detected by M ∆ h e . Proof.
Table 3 shows that the Massey product h ∆ h e , h , h g i equals theelement M ∆ h e . Now apply the Moss Convergence Theorem 2.16, using thatTable 17 shows that ∆ h e detects τ ηκ . (cid:3) Lemma . There exists an element α in π , that is detected by h Q + h n such that h c detects the Toda bracket h τ α, ν , η i . Proof.
A consequence of the proof of Lemma 5.48 is that there exists α in π , that is detected by h Q + h n such that the product τ ν α is zero. There-fore, h c detects the Toda bracket h τ α, ν , η i because of the Adams differential d ( h c ) = τ h h h Q . (cid:3) HAPTER 7
Hidden extensions
In this chapter, we will discuss hidden extensions in the E ∞ -page of the Adamsspectral sequence. We methodically explore hidden extensions by τ , 2, η , and ν ,and we study other miscellaneous hidden extensions that are relevant for specificpurposes. τ extensions In order to study hidden τ extensions, we will use the long exact sequence(7.1) · · · / / π p,q +1 τ / / π p,q / / π p,q Cτ / / π p − ,q +1 τ / / π p − ,q / / · · · extensively. This sequence governs hidden τ extensions in the following sense. Anelement α in π p,q is divisible by τ if and only if it maps to zero in π p,q Cτ , and anelement α in π p − ,q +1 supports a τ extension if and only if it is not in the imageof π p,q Cτ . Therefore, we need to study the maps π ∗ , ∗ → π ∗ , ∗ Cτ and π ∗ , ∗ Cτ → π ∗− , ∗ +1 induced by inclusion of the bottom cell into Cτ and by projection from Cτ to the top cell.The E ∞ -pages of the Adams spectral sequences for S , and Cτ give associatedgraded objects for the homotopy groups that are the sources and targets of thesemaps. Naturality of the Adams spectral sequence induces maps on associatedgraded objects.These maps on associated graded objects often detect the values of the mapson homotopy groups. For example, the element h in the Adams spectral sequencefor the sphere is mapped to the element h in the Adams spectral sequence for Cτ .In homotopy groups, this means that inclusion of the bottom cell into Cτ takes theelement 2 in π , to the element 2 in π , Cτ .On the other side, the element h in the Adams spectral sequence for Cτ ismapped to the element h in Adams spectral sequence for the sphere. In homotopygroups, this means that projection from Cτ to the top cell takes the element { h } in π , Cτ to the element η in π , .However, some values of the maps on homotopy groups can be hidden in themap of associated graded objects. This situation is rare in low stems but becomesmore and more common in higher stems. The first such example occurs in the30-stem. The element ∆ h is a permanent cycle in the Adams spectral sequencefor Cτ , so { ∆ h } is an element in π , Cτ . Now ∆ h maps to zero in the E ∞ -pageof the Adams spectral sequence for the sphere, but { ∆ h } does not map to zero in π , . In fact { ∆ h } maps to ηκ , which is detected by h d . This demonstratesthat projection from Cτ to the top cell has a hidden value.We refer the reader to Section 2.1 for a precise discussion of these issues. Theorem .
590 7. HIDDEN EXTENSIONS (1) Through the 90-stem, Table 12 lists all hidden values of inclusion of thebottom cell into Cτ , except that:(a) If h x , does not survive, then τ h x , maps to ∆ e + τ ∆ h e g .(b) If h h g or h h g + h f does not survive, then τ h h g or τ h h g + τ h f maps to ∆ e + τ ∆ h e g .(c) If τ ∆ h H survives, then it maps to ∆ h B .(2) Through the 90-stem, Table 13 lists all hidden values of projection from Cτ to the top cell, except that:(a) If τ h g + τ h e does not survive, then τ h g maps to τ (∆ e + C ) g .(b) If x , does not survive, then x , maps to ∆ h j .(c) If h x , does not survive, then h x , maps to τ M h g .(d) If h x , survives, then ∆ e + τ ∆ h e g maps to M ∆ h d .(e) If h h g or h h g + h f does not survive, then h h g maps to τ M h g .(f ) If x , does not survive, then x , maps to M ∆ h e .(g) If τ ∆ h H does not survive, then ∆ h B maps to M ∆ h e . Proof.
The values of inclusion of the bottom cell and projection to the topcell are almost entirely determined by inspection of Adams E ∞ -pages. Taking intoaccount the multiplicative structure, there are no other combinatorial possibilities.For example, consider the exact sequence π , → π , Cτ → π , . In the Adams E ∞ -page for Cτ , h and ∆ h are the only two elements in the 30-stem with weight 16. In the Adams E ∞ -page for the sphere, h is the only elementin the 30-stem with weight 16, and h d is the only element in the 29-stem withweight 17. The only possibility is that h maps to h under inclusion of the bottomcell, and ∆ h maps to h d under projection to the top cell.One case, given below in Lemma 7.7, requires a more complicated argument. (cid:3) Remark . Through the 90-stem, inclusion of the bottom cell into Cτ hasonly one hidden value with target indeterminacy. Namely, h c A ′ is the hiddenvalue of h gB , with target indeterminacy generated by ∆ j . Through the 90-stem,projection from Cτ to the top cell has no hidden values with target indeterminacy. Remark . Through the 90-stem, inclusion of the bottom cell into Cτ hasno crossing values. On the other hand, projection from Cτ to the top cell does havecrossing values in this range. These occurrences are described in the fourth columnof Table 13. Each can be verified by direct inspection. Theorem . Through the 90-stem, Table 14 lists all hidden τ extensions in C -motivic stable homotopy groups, except that:(1) if ∆ h n is not hit by a differential, then there is a hidden τ extensionfrom τ (∆ e + C ) g to ∆ h n .(2) if M ∆ h d is not hit by a differential, then there is a hidden τ extensionfrom ∆ h j to M ∆ h d .(3) if M ∆ h d is not hit by a differential, then there is a hidden τ extensionfrom τ M h g to M ∆ h d .In this range, the only crossing extension is: .2. HIDDEN 2 EXTENSIONS 61 (1) the hidden τ extension from h h c to h h D , and the not hidden τ extension on τ h Q . Proof.
Almost all of these hidden τ extensions follow immediately from thevalues of the maps in the long exact sequence (7.1) given in Tables 12 and 13.For example, consider the element P d in the Adams E ∞ -page for the sphere,which belongs to the 22-stem with weight 12. Now π , Cτ is zero because thereare no elements in that degree in the Adams E ∞ -page for Cτ , so inclusion ofthe bottom cell takes { P d } to zero. Therefore, { P d } must be in the image ofmultiplication by τ . The only possibility is that there is a hidden τ extension from c d to P d . (cid:3) Remark . A straightforward analysis of the sequence (7.1) shows that∆ h n is not hit by a differential if and only if the possible τ extension on τ (∆ e + C ) g occurs. Thus this uncertain hidden τ extension is entirely determined by acorresponding uncertainty in a value of an Adams differential. Remark . If M ∆ h d and M ∆ h d are not hit by differentials, then astraightforward analysis of the sequence (7.1) shows that the possible τ extensionson ∆ h j and τ M h g must occur. Thus these uncertainties are entirely determinedby corresponding uncertainties in values of the Adams differentials. Lemma . (1) The element h h (∆ e + C ) + τ h C ′′ maps to h c Q under inclusion ofthe bottom cell into Cτ .(2) There is a hidden τ extension from d e to h h (∆ e + C ) . Proof.
Consider the exact sequence π , → π , Cτ → π , . For combi-natorial reasons, one of the following two possibilities must occur:(a) the element h h (∆ e + C ) + τ h C ′′ maps to h c Q under inclusion of thebottom cell into Cτ , and there is a hidden τ extension from d e to h h (∆ e + C ).(b) the element h h (∆ e + C ) maps to h c Q under inclusion of the bottom cellinto Cτ , and there is a hidden τ extension from d e to h h (∆ e + C )+ τ h C ′′ .We will show that there cannot be a hidden τ extension from d e to h h (∆ e + C ) + τ h C ′′ .Lemma 7.157 shows that τ ν { d e } equals τ ησ { k } . Since there is no hidden τ extension on h k , there must exist an element α in { k } such that τ ηα = 0.Therefore, τ ν { d e } must be zero.If there were a τ extension from d e to h h (∆ e + C )+ τ h C ′′ , then τ ν { d e } would be detected by h · ( h h (∆ e + C ) + τ h C ′′ ) = τ h C ′′ , and in particular would be non-zero. (cid:3) extensions Theorem . Table 15 lists some hidden extensions by . Proof.
Many of the hidden extensions follow by comparison to Cτ . For ex-ample, there is a hidden 2 extension from h h g to h c d in the Adams spectralsequence for Cτ . Pulling back along inclusion of the bottom cell into Cτ , there must also be a hidden 2 extension from h h g to h c d in the Adams spectralsequence for the sphere. This type of argument is indicated by the notation Cτ inthe fourth column of Table 15.Next, Table 14 shows a hidden τ extension from h c d to P h d . Therefore,there is also a hidden 2 extension from τ h h g to P h d . This type of argument isindicated by the notation τ in the fourth column of Table 15.Many cases require more complicated arguments. In stems up to approximatelydimension 62, see [ , Section 4.2.2 and Tables 27–28] [ ], and [ ]. The higher-dimensional cases are handled in the following lemmas. (cid:3) Remark . Through the 90-stem, there are no crossing 2 extensions.
Remark . The hidden 2 extension from h h g to τ gn is proved in [ ],which relies on the “ R P ∞ -method” to establish a hidden σ extension from τ h d to∆ h c and a hidden η extension from τ h g to ∆ h c . We now have easier proofsfor these η and σ extensions, using the hidden τ extension from h g to ∆ h c given in Table 14, as well as the relation h d = h g . Remark . Comparison to synthetic homotopy gives additional informationabout some possible hidden 2 extensions, including:(1) there is a hidden 2 extension from h h i to τ e g .(2) there is no hidden 2 extension from P x , to M ∆ h d .See [ ] for more details. We are grateful to John Rognes for pointing out a mistakein [ , Lemma 4.56 and Table 27] concerning the hidden 2 extension on h h i .Lemma 7.19 shows that the extension occurs but does not determine its targetprecisely. Remark . The first correct proof of the relation 2 θ = 0 appeared in [ ].Earlier claims in [ ] and [ ] were based upon a mistaken understanding of theToda bracket h σ , , θ i . See [ , Table 23] for the correct value of this bracket. Remark . If M ∆ h d is non-zero in the E ∞ -page, then there is a hidden τ extension from τ M h g to M ∆ h d . This implies that there must be a hidden 2extension from τ M g to M ∆ h d . Theorem . Table 16 lists all unknown hidden extensions, through the -stem. Proof.
Many possibilities are eliminated by comparison to Cτ , to tmf , or to mmf . For example, there cannot be a hidden 2 extension from h h to τ h g bycomparison to Cτ .Many additional possibilities are eliminated by consideration of other parts ofthe multiplicative structure. For example, there cannot be a hidden 2 extensionfrom P h h to τ g because τ g supports an h extension and 2 η equals zero.Several cases are a direct consequence of Proposition 7.17.Some possibilities are eliminated by more complicated arguments. These casesare handled in the following lemmas. (cid:3) Remark . The element τ h g detects the product κθ , so it cannot supporta hidden 2 extension since 2 θ is zero. If τ h g + τ h e survives, then there is ahidden τ extension from τ (∆ e + C ) g to ∆ h n . Then there could not be a hidden2 extension from h g to τ (∆ e + C ) g . .2. HIDDEN 2 EXTENSIONS 63 Remark . If M ∆ h d is not zero in the E ∞ -page, then M ∆ h d supportsan h multiplication, and there cannot be a hidden 2 extension from P x , to M ∆ h d . Proposition . Suppose that α and τ ηα are both zero. Then h α, , θ i is zero. Proof.
Consider the shuffle2 h α, , θ i = h , α, i θ . Since 2 θ is zero, this expression has no indeterminacy. Corollary 6.2 implies thatit equals τ ηαθ , which is zero by assumption. (cid:3) Remark . Proposition 7.17 eliminates possible hidden 2 extensions onseveral elements, including h h , h h h , h h , h c , h h h , h h i , and h h g . Lemma . There is a hidden extension from h h i to either τ M P h orto τ e g . Proof.
Table 2 shows that h h i maps to ∆ h in the homotopy of tmf . Theelement ∆ h supports a hidden 2 extension, so h h i must support a hidden 2extension as well. (cid:3) Lemma . (1) There is a hidden extension from τ h H to τ h (∆ e + C ) .(2) There is no hidden extension on τ X + τ C ′ .(3) There is a hidden extension from τ h h H to τ h h (∆ e + C ) . Proof.
Table 17 shows that there is an η extension from τ h H to h Q .Let α be any element of π , that is detected by τ h H . Then τ η α is non-zeroand detected by τ h h Q . Note that τ h h Q cannot be the target of a hidden 2extension because there are no possibilities.If 2 α were zero, then we would have the shuffling relation τ η α = h , η, i α = 2 h η, , α i . But this would contradict the previous paragraph.We now know that 2 α must be non-zero for every possible choice of α . Theonly possibility is that there is a hidden 2 extension from τ h H to τ h (∆ e + C ),and that there is no hidden 2 extension on τ X + τ C ′ . This establishes the firsttwo parts.The third part follows immediately from the first part by multiplication by h . (cid:3) Lemma . There is a hidden extension from h h to τ h h . Proof.
Table 11 shows that h h detects h η, , θ i . Now shuffle to obtain2 h η, , θ i = h , η, i θ = τ η θ . (cid:3) Lemma . There is no hidden extension on ∆ h . Proof.
Table 11 shows that ∆ h detects the Toda bracket h η , θ , η , θ i . Wehave 2 h η , θ , η , θ i ⊆ hh , η , θ i , η , θ i . Table 11 shows that νθ = h , η, ηθ i = h , η , θ i , so we must compute h νθ , η , θ i .This bracket contains ν h θ , η , θ i , which equals zero by Lemma 6.14. There-fore, we only need to compute the indeterminacy of h νθ , η , θ i .The only possible non-zero element in the indeterminacy is the product θ { t } .Table 11 shows that { t } = h ν, η, ηθ i . Now θ { t } = h ν, η, ηθ i θ = ν h η, ηθ , θ i . This last expression is well-defined because θ is zero [ ], and it must be zerobecause π , consists entirely of multiples of η . (cid:3) Lemma . There is no hidden extension on h Q + h D . Proof.
By comparison to the homotopy of Cτ , there is no hidden extensionwith value h A ′ . Table 17 shows that τ ∆ h e g supports a hidden η extension.Therefore, it cannot be the target of a 2 extension. (cid:3) Lemma . There is no hidden extension on h A ′ . Proof.
Table 11 shows that h A ′ detects the Toda bracket h σ, κ, τ ηθ . i . Shuf-fle to obtain h σ, κ, τ ηθ . i σ h κ, τ ηθ . , i . The bracket h κ, τ ηθ . , i is zero because it is contained in π , = 0. (cid:3) Lemma . There is no hidden extension on p ′ . Proof.
Table 21 shows that p ′ detects the product σθ , and 2 θ is alreadyknown to be zero [ ]. (cid:3) Lemma . There is no hidden extension on τ h D ′ . Proof.
Table 11 shows that τ h D ′ detects the Toda bracket h η, ν, τ θ . κ i .Now shuffle to obtain 2 h η, ν, τ θ . κ i = h , η, ν i τ θ . κ, which equals zero because h , η, ν i is contained in in π , = 0. (cid:3) Lemma . There is no hidden extension on h h h . Proof.
Table 11 shows that h h detects the Toda bracket h η, , θ i . Let α be an element of this bracket. Then h h h detects σα , and2 σα = 2 σ h η, , θ i = σ h , η, i θ = τ η σθ . Table 11 also shows that h c detects the Toda bracket h ǫ, , θ i . Let β be anelement of this bracket. As in the proof of Lemma 7.28, we compute that 2 β equals τ ηǫθ .Now consider the element σα + β , which is also detected by h h h . Then2( σα + β ) = τ η σθ + τ ηǫθ = τ ν θ , using Toda’s relation η σ + ν = ηǫ [ ]. .2. HIDDEN 2 EXTENSIONS 65 Table 19 shows that there is a hidden ν extension from h h to τ h Q . There-fore, τ h Q detects ν θ .This does not yet imply that ν θ is zero, because ν θ + η { τ Q + τ n } mightbe detected h A ′ or P h h j in higher filtration. However, h A ′ does not supporta hidden ν extension by Lemma 7.113. Also, Table 2 shows that P h h j mapsnon-trivially to tmf , while ν θ + η { τ Q + τ n } maps to zero. This is enough toconclude that ν θ is zero.We have now shown that 2( σα + β ) is zero in π , . Since h h h detects σα + β , it follows that h h h does not support a hidden 2 extension. (cid:3) Lemma . There is a hidden extension from h c to τ h p ′ . Proof.
Table 11 shows that h c detects the Toda bracket h ǫ, , θ i . Nowshuffle to obtain 2 h ǫ, , θ i = h , ǫ, i θ = τ ηǫθ . Finally, τ ηǫθ is detected by τ h p ′ because of the relation h p ′ = h h c . (cid:3) Lemma . There is no hidden extension on τ h p . Proof.
Lemma 6.18 shows that τ h p detects h η, ν, α i for some α detected by τ h C ′ . Now shuffle to obtain 2 h η, ν, α i = h , η, ν i α, which is zero because h , η, ν i is contained in π , = 0. (cid:3) Lemma . There is a hidden extension from h H to τ M h g . Proof.
Table 19 shows that there are hidden ν extensions from h H to h C ′′ ,and from τ M h g to M h d . Table 15 shows that there is also a hidden 2 extensionfrom h C ′′ to M h d . The only possibility is that there must also be a hidden 2extension on h h H . (cid:3) Lemma . There is no hidden extension on P h h . Proof.
Table 11 shows that
P h h detects the Toda bracket h µ , , θ i . Shuffleto obtain 2 h µ , , θ i = h , µ , i θ = τ ηµ θ . Table 11 also shows that µ is contained in the Toda bracket h η, , σ i . Shuffleagain to obtain τ ηµ θ = h η, , σ i τ ηθ = τ η h , σ, θ i . Table 11 shows that h h h detects h , σ, θ i .By inspection, the product η { h h h } can only be detected by ∆ h h c .However, this cannot occur by comparison to Cτ . Therefore, η { h h h } , and also τ η { h h h } , must be zero. (cid:3) Lemma . There is no hidden extension on h Q + h D . Proof.
Table 11 shows that the element h Q + h D detects the Toda bracket h σ , , { t } , τ κ i . Consider the relation2 h σ , , { t } , τ κ i ⊆ hh , σ , i , { t } , τ κ i . Corollary 6.2 shows that the Toda bracket h , σ , i contains zero since ησ is zero.Therefore, it consists of even multiples of ρ ; let 2 kρ be any such element in π , . The Toda bracket h kρ , { t } , τ κ i contains kρ h , { t } , τ κ i , which equals zeroas discussed in the proof of Lemma 6.19. Moreover, its indeterminacy is equal to τ κ · π , , which is detected in Adams filtration at least 12. This implies that h kρ , { t } , τ κ i is detected in Adams filtration at least 12, and that the target of ahidden 2 extension on h Q + h D must have Adams filtration at least 12.The remaining possible targets with Adams filtration at least 12 are eliminatedby comparison to Cτ or to mmf . (cid:3) Remark . The proof of Lemma 7.32 might be simplified by consideringthe shuffle 2 h σ , , { t } , τ κ i = h , σ , , { t }i τ κ. However, the latter four-fold bracket may not exist, since both three-fold subbrack-ets have indeterminacy. See [ ] for a discussion of the analogous difficulty withMassey products. Lemma . There is no hidden extension on h Q . Proof.
The element τ h Q detects ν { τ Q + τ n } , so it cannot support ahidden 2 extension. This rules out all possible 2 extensions on h Q . (cid:3) Lemma . There is no hidden extension on h h D . Proof.
Table 14 shows that there is a hidden τ extension from h h c to h h D . Therefore, h h D detects either τ ηǫη or τ ηǫη + ν { τ Q + τ n } , becauseof the presence of τ h Q in higher filtration. In either case, h h D cannot supporta hidden 2 extension. (cid:3) Lemma . There is a hidden extension from h ( τ Q + τ n ) to τ x , . Proof.
Table 21 shows that τ x , detects τ κ θ . Table 11 shows that θ equals the Toda bracket h σ , , σ , i .Now consider the shuffle τ κ θ = τ κ h σ , , σ , i = h τ κ , σ , , σ i . Lemma 6.11 shows that the latter bracket is well-defined. This implies that τ x , isthe target of a hidden 2 extension, and h ( τ Q + τ n ) is the only possible source. (cid:3) Lemma . There is no hidden extension on h d . Proof.
Table 11 shows that h d detects the Toda bracket h κ, , θ i . Nowshuffle to obtain 2 h κ, , θ i = h , κ, i θ = τ ηκθ . Lemma 7.156 shows that this product equals τ ησ θ , which equals zero because ησ is zero. (cid:3) Lemma . There is a hidden extension from e A ′ to M ∆ h h . Proof.
Let α be an element of π , that is detected by x , . Table 17 showsthat there is a hidden η extension from x , to M ∆ h h , so τ η α is detected by τ M ∆ h h . Now shuffle to obtain τ η α = h , η, i α = 2 h η, , α i . This shows that τ M ∆ h h must be the target of a hidden 2 extension. .2. HIDDEN 2 EXTENSIONS 67 Moreover, the source of this hidden 2 extension must be in Adams filtration atleast 10, since the Adams differential d ( τ x , ) = h x , implies that h η, , α i isdetected by h x , = 0 in filtration 9. The only possible source is e A ′ . (cid:3) Lemma . There is no hidden extension on h h h . Proof.
Table 11 shows that h h h detects the Toda bracket h η , , θ i . Nowshuffle to obtain 2 h η , , θ i = h , η , i θ , which equals τ ηη θ by Table 11. We will show that this product is zero.There are several elements in the Adams E ∞ -page that might detect η θ . Thepossibilities h h d and x , are ruled out by comparison to Cτ . The possibility τ e A ′ is ruled out because Table 15 shows that e A ′ supports a hidden 2 extension.Two possibilities remain. If η θ is detected by τ M ∆ h h , then τ ηη θ mustbe zero because there are no elements in sufficiently high Adams filtration.Finally, suppose that η θ is detected by τ h x , . Let α be an element of π , that is detected by h x , . If η θ + τ ηα is not zero, then it is detected inhigher filtration. It cannot be detected by τ e A ′ because of the hidden 2 extensionon e A ′ . If it is detected by τ M ∆ h h , then we may change the choice of α toensure that η θ + τ ηα is zero.We have now shown that τ ηη θ equals τ η α . Shuffle to obtain τ η α = τ α h , η, i = h α, , η i τ. Table 11 shows that h α, , η i is detected by h h x , , and Lemma 7.40 showsthat this element does not support a hidden 2 extension. Therefore, h α, , η i τ iszero. (cid:3) Lemma . There is no hidden extension on h h x , . Proof.
Let α be an element of π , that is detected by h A . Then να isdetected by h h x , .Table 21 shows that h h A detects σ θ . Then 2 α + σ θ equals zero, or isdetected by x , . This implies that 2 να either equals zero, or is detected by h x , .However, there cannot be a hidden 2 extension from h h x , to h x , bycomparison to Cτ . (cid:3) Lemma . There is no hidden extension on P h c . Proof.
Table 21 shows that
P h c detects the product ρ η . Table 11 showsthat η is contained in the Toda bracket h η, , θ i . Now shuffle to obtain2 ρ η = 2 ρ h η, , θ i = ρ θ h , η, i , which equals τ η ρ θ by Table 11.Table 21 shows that ρ θ is detected by either h x , or τ m . First supposethat it is detected by h x , . Table 15 shows that h x , is the target of a 2extension. Then ρ θ equals 2 α modulo higher filtration. In any case, τ η ρ θ iszero.Next suppose that ρ θ is detected by τ m . Then ρ θ equals τ α modulohigher filtration for some element α detected by m . Table 14 shows that there isa hidden τ extension from h m to M ∆ h h . This implies that τ ηα is detected by M ∆ h h . Finally, τ η α = τ η ρ θ must be zero. (cid:3) Lemma . There is no hidden extension on ∆ B . Proof.
Table 17 shows that there is a hidden η extension from h h h to τ ∆ B . Therefore, τ ∆ B cannot be the source of a hidden 2 extension, so therecannot be a hidden 2 extension from ∆ B to τ M e . (cid:3) Lemma . There is no hidden extension on h h g . Proof.
The element h h g equals h h d , so it detects ν { h d } . (cid:3) Lemma . There is no hidden extension on τ h h g . Proof. If τ h g + τ h e is a permanent cycle, then τ h h g detects a multiple of η and cannot support a hidden 2 extension. However, we need a more complicatedargument because we do not know if τ h g + τ h e survives.The element τ h h g detects η κ . Table 11 shows that η is contained in theToda bracket h η, , θ i . Shuffle to obtain2 η κ = 2 h η, , θ i κ = h , η, i θ κ = τ η θ κ. The product θ κ is detected by τ h g .There are several possible values for η { h g } , but they are either ruled out bycomparison to Cτ , or they are multiples of h . In all cases, η { h g } must be zero.This implies that τ η θ κ is also zero. (cid:3) Lemma . There is a hidden extension from τ h f to τ h h Q . Proof.
Table 19 shows that τ h Q detects ν θ . The Moss Convergence The-orem 2.16 implies that τ h h Q detects the Toda bracket h , η , τ ν θ i , using thedifferential d ( h c ) = τ h h Q .On the other hand, this bracket contains h , η , ν i τ νθ . The bracket h , η , ν i in π , must contain zero by comparison to tmf .We have now shown that τ h h Q detects a linear combination of a multipleof 2 and a multiple of τ ν θ . However, τ ν θ · π , is zero, so τ h h Q detects amultiple of 2. The only possibility is that there is a hidden 2 extension from τ h f to τ h h Q . (cid:3) Lemma . There is no hidden extension on τ h h Q . Proof.
There cannot be a hidden 2 extension from τ h h Q to τ P h x , because there is no hidden τ extension from h h h Q to P h x , .Table 17 shows that τ M g supports a hidden η extension. Therefore, it cannotbe the target of a hidden 2 extension. (cid:3) Lemma . Neither h c d nor P h d support hidden extensions. Proof.
Table 17 shows that both elements are targets of hidden η extensions. (cid:3) Lemma . There is no hidden extension on h h c . Proof.
Table 21 shows that h h c detects the product σ { h h h } , and theelement h h h does not support a hidden 2 extension by Lemma 7.39. (cid:3) Lemma . There is no hidden extension on τ h gC ′ . .3. HIDDEN η EXTENSIONS 69
Proof.
The possible target τ e m is ruled out by comparison to mmf . Thepossible target P h h is ruled out by comparison to Cτ .It remains to eliminate the possible target τ M h g . Table 14 shows thatthere are hidden τ extensions from τ h gC ′ and τ M h g to ∆ h d and M ∆ h e respectively. However, there is no hidden 2 extension from ∆ h d to M ∆ h e , sothere cannot be a 2 extension from τ h gC ′ to τ M h g . (cid:3) Lemma . There is no hidden extension on h c . Proof.
Table 11 shows that the Toda bracket h τ { h Q + h n } , ν , η i is de-tected by h c . Shuffle to obtain h τ { h Q + h n } , ν , η i τ { h Q + h n }h ν , η, i . These expressions have no indeterminacy because τ { h Q + h n } does not supporta 2 extension. Finally, the bracket h ν , η, i contains zero by comparison to tmf . (cid:3) Lemma . If τ ∆ h H survives, then it supports a hidden extension toeither τ ∆ h d or to τ ∆ c g . Proof.
Suppose that τ ∆ h H survives. Let α be an element of π , thatis detected by τ ∆ h H . Comparison to Cτ shows that ηα is detected by ∆ f or ∆ f + τ ∆ g g , depending only on which one survives. Then τ ∆ h f detects τ η α .If 2 α were zero, then we could shuffle to obtain τ η α = h , η, i α = 2 h η, , α i . Considering the Massey product h h , h , τ ∆ h H i , the Moss Convergence Theorem2.16 would imply that h η, , α i is detected in filtration at least 11. But there areno possible 2 extensions whose source is in filtration at least 11 and whose targetis τ ∆ h f . (cid:3) Lemma . There is no hidden extension on P h c . Proof.
Table 21 shows that P h c detects the product ρ η . Table 11 showsthat η is contained in the Toda bracket h η, , θ i . Shuffle to obtain2 ρ η = 2 ρ h η, , θ i = h , η, i ρ θ = τ η ρ θ . The product ρ θ is detected in Adams filtration at least 13, and then τ η ρ θ isdetected in filtration at least 16. This rules out all possible targets for a hidden 2extension on P h c . (cid:3) Lemma . There is no hidden extension on M . Proof.
Table 21 shows that M detects θ . . Graded commutativity impliesthat 2 θ . is zero. (cid:3) η extensions Theorem . Table 17 lists some hidden extensions by η . Proof.
Many of the hidden extensions follow by comparison to Cτ . For exam-ple, there is a hidden η extension from τ h g to c d in the Adams spectral sequencefor Cτ . Pulling back along inclusion of the bottom cell into Cτ , there must alsobe a hidden η extension from τ h g to c d in the Adams spectral sequence for the sphere. This type of argument is indicated by the notation Cτ in the fourth columnof Table 17.Next, Table 14 shows a hidden τ extension from c d to P d . Therefore, thereis also a hidden η extension from τ h g to P d . This type of argument is indicatedby the notation τ in the fourth column of Table 17.The proofs of several of the extensions in Table 17 rely on analogous extensionsin mmf . Extensions in mmf have not been rigorously analyzed [ ]. However, thespecific extensions from mmf that we need are easily deduced from extensions in tmf , together with the multiplicative structure. For example, there is a hidden η extension in tmf from an to τ d . Therefore, there is a hidden η extension in mmf from ang to τ d g , and also a hidden η extension from ∆ h e to τ d e in thehomotopy groups of the sphere spectrum. Note that mmf really is required here,since ang and d g equal zero in the homotopy of tmf .Many cases require more complicated arguments. In stems up to approxi-mately dimension 62, see [ , Section 4.2.3 and Tables 29–30] and [ ]. Thehigher-dimensional cases are handled in the following lemmas. (cid:3) Remark . The hidden η extension from τ C to τ gn is proved in [ ],which relies on the “ R P ∞ -method” to establish a hidden σ extension from τ h d to ∆ h c and a hidden η extension from τ h g to ∆ h c . We now have easier proofsfor these η and σ extensions, using the hidden τ extension from h g to ∆ h c givenin Table 14, as well as the relation h d = h g . Remark . By comparison to Cτ , h h h must support a hidden η ex-tension. The only possible targets are τ ∆ B and ∆ n . If ∆ h n is not hit by adifferential, then the target must be τ ∆ B . Remark . If x , survives, then there is a hidden τ extension from ∆ h j to M ∆ h d . It follows that there must be a hidden η extension from τ ∆ j + τ gC ′ to M ∆ h d . Remark . The last column of Table 17 indicates the crossing η extensions. Theorem . Table 18 lists all unknown hidden η extensions, through the -stem. Proof.
Many possible extensions can be eliminated by comparison to Cτ , to tmf , or to mmf . For example, there cannot be a hidden η extension from τ M d to τ g because τ g maps to a non-zero element in π tmf that is not divisible by η .Other possibilities are eliminated by consideration of other parts of the multi-plicative structure. For example, there cannot be a hidden η extension whose targetsupports a multiplication by 2, since 2 η equals zero.Many cases are eliminated by more complicated arguments. These are handledin the following lemmas. (cid:3) Remark . If τ h g + τ h e survives, then τ (∆ e + C ) g supports a hidden τ extension. It follows that h h h cannot support a hidden η extension because τ η { h h h } = 4 ν { h h h } must be zero. Remark . If τ h g + τ h e survives, then there is a hidden τ extension from τ (∆ e + C ) g to ∆ h n . Then the possible extension from τ gD to τ (∆ e + C ) g occurs if and only if the possible extension from τ gD to ∆ h n occurs. .3. HIDDEN η EXTENSIONS 71
Remark . If τ h g + τ h e survives, then τ h h g + τ h f is a multiple of h . This implies that τ h h g + τ h f cannot support a hidden η extension. On theother hand, if h h g or h h g + h f does not survive, then τ h h g + τ h f mapsto ∆ e + τ ∆ h e g in the Adams E ∞ -page for Cτ . This element supports an h multiplication, so τ h h g + τ h f must support a hidden η extension. Remark . If h x , does not survive, then τ h x , maps to ∆ e + τ ∆ h e g in the Adams E ∞ -page for Cτ . This element supports an h multiplica-tion, so τ h x , must support a hidden η extension. Lemma . There is no hidden η extension on τ h Q . Proof.
There cannot be a hidden η extension from τ h Q to τ ∆ h d g bycomparison to tmf . It remains to show that there cannot be a hidden η extensionfrom τ h Q to τ M d .Note that h d Q = τ d g , so κ { h Q } is detected by τ d g . Therefore, κ { h Q } + τ κ κ is detected in higher filtration. The only possibility is τ e gm ,but that cannot occur by comparison to mmf . Therefore, κ { h Q } + τ κ κ is zero.Now τ ηκ κ is zero because τ ηκ κ cannot be detected by ∆ h d by comparisonto tmf . Therefore, ηκ { h Q } is zero, so τ ηκ { h Q } is also zero.On the other hand, τ κ { M d } is non-zero because it is detected by τ M d .Therefore τ η { h Q } cannot be detected by τ M d . (cid:3) Lemma . There is no hidden η extension on τ h h . Proof.
Table 15 shows that τ h h is the target of a hidden 2 extension. (cid:3) Lemma . There is a hidden η extension from τ h X to τ M h g . Proof.
Table 17 shows that there is a hidden η extension from τ h X to c Q . Since c Q does not support a hidden τ extension, there exists an element β in π , that is detected by c Q such that τ β = 0.Projection from Cτ to the top cell takes c Q and P ( A + A ′ ) to c Q and τ M h h g respectively. Since h · c Q = P ( A + A ′ ) in the Adams spectral sequencefor Cτ , it follows that νβ is non-zero and detected by τ M h h g .Let α be an element of π , that is detected by τ X + τ C ′ , and consider thesum η α + β . Both terms are detected by c Q , but the sum could be detected inhigher filtration. In fact, the sum is non-zero because ν ( η α + β ) is non-zero.It follows that η α + β is detected by τ M h g , and that τ η α is detected by τ M h g . (cid:3) Lemma . There is no hidden η extension on τ h h . Proof.
The element τ η η is detected by τ h h . Table 11 shows that η iscontained in the Toda bracket h η, , θ i Now shuffle to obtain η · τ η η = 4 νη = 4 ν h η, , θ i = 4 h ν, η, i θ , which equals zero because 2 θ is zero. (cid:3) Lemma . There is no hidden η extension from τ ∆ h to h A ′ . Proof.
Table 19 shows that h A ′ supports a hidden ν extension, so it cannotbe the target of a hidden η extension. (cid:3) Lemma . There is no hidden η extension on τ h Q . Proof.
Table 19 shows that τ h Q is the target of a hidden ν extension.Therefore, it cannot be the source of a hidden η extension. (cid:3) Lemma . There is a hidden η extension from h A ′ to h (∆ e + C ) . Proof.
Comparison to Cτ shows that there is a hidden η extension from h A ′ to either τ h C ′ + h (∆ e + C ) or h (∆ e + C ). Table 19 shows that τ h C ′ + h (∆ e + C ) supports a hidden ν extension. Therefore, it cannot be the target ofa hidden η extension. (cid:3) Lemma . There is a hidden η extension from h h to τ h h Q . Proof.
Table 3 gives the Massey product h h = h h , h , h i . Therefore, h τ h Q , h , h i = { τ h h Q , τ h h Q + τ h h H } . Table 19 shows that there is a hidden ν extension from h h to τ h Q , so ν θ isdetected by τ h Q . Therefore, the Toda bracket h ν θ , , η i is detected by τ h h Q or by τ h h Q + τ h h H .Now h ν θ , , η i contains ν h θ , , η i . This expression equals νθ h , η, ν i , whichequals zero because h , η, ν i is contained in π , = 0.We now know that h ν θ , , η i equals its own determinacy, so τ h h Q or τ h h Q + τ h h H detects a multiple of η . The only possibility is that thereis a hidden η extension on h h .The target of this extension cannot be τ h h Q + τ h h H by comparison to Cτ . (cid:3) Lemma . There is a hidden η extension from τ h h H to h Q . Proof.
Table 17 shows that there is a hidden η extension from τ h H to h Q .Now multiply by h . (cid:3) Lemma . (1) There is no hidden η extension on h h (∆ e + C ) .(2) There is no hidden η extension on τ h C ′′ + h h (∆ e + C ) . Proof.
The element τ M h g is the only possible target for such hidden η extensions. However, Table 19 shows that there is a hidden ν extension from τ M h g to M h d . (cid:3) Lemma . There is no hidden η extension on h h h . Proof.
There are several possible targets for a hidden η extension on h h h .The element τ ∆ h g is ruled out because it supports an h extension. The element∆ h c is ruled out by comparison to Cτ . The elements τ h Q and τ d Q areruled out because Table 17 shows that they are targets of hidden η extensions from τ h h H and τ h D ′ respectively.The only remaining possibility is τ l . This case is more complicated.Table 11 shows that h h h detects the Toda bracket h σ, , θ i . Now shuffleto obtain η h σ, , θ i = h η, σ, i θ . Table 11 shows that h η, σ, i contains µ and has indeterminacy generated by τ η σ and τ ηǫ . Thus the expression h η, σ, i θ contains at most four elements.The product µ θ is detected in filtration at least 8, so it is not detected by τ l .The product ( µ + τ η σ ) θ is detected by τ h p ′ because Table 21 shows that there .3. HIDDEN η EXTENSIONS 73 is a hidden σ extension from h to p ′ . The product ( µ + τ ηǫ ) θ is also detected by τ h p ′ = τ h h c . Finally, the product ( µ + + τ η σ + τ ηǫ ) θ equals ( µ + τ ν ) θ ,which also must be detected in filtration at least 8. (cid:3) Lemma . There is no hidden η extension on h Q . Proof.
There cannot be a hidden η extension on τ h Q because it is a multipleof h . Therefore, the possible targets for an η extension on h Q must be annihilatedby τ .The element h h H cannot be the target because Table 14 shows that itsupports a hidden τ extension. The element τ M h g cannot be the target becauseTable 19 shows that it supports a hidden ν extension to M h d . (cid:3) Lemma . There is no hidden η extension on τ h p ′ . Proof.
The element τ h p ′ detects τ η σθ because Table 21 shows that thereis a hidden σ extension from h to p ′ . Then τ η σθ is zero since τ η σ is zero. (cid:3) Lemma . There is no hidden η extension on h H . Proof.
Table 21 shows that
M d detects the product κθ . . Then Table 11shows that h H detects the Toda bracket h ν, ǫ, κθ . i . Now shuffle to obtain η h ν, ǫ, κθ . i = h η, ν, ǫ i κθ . , which is zero because h η, ν, ǫ i is contained in π , = 0. (cid:3) Lemma . There is a hidden η extension from τ h h c to τ h Q . Proof.
The hidden τ extension from h h c to h d D implies that τ h h c must support a hidden η extension. However, this hidden τ extension crosses the τ extension from τ h Q to τ h Q . Therefore, the target of the hidden η extensionis either τ h Q or h d D .The element τ h h c detects the product τ η ǫ , so we want to compute τ ηη ǫ .Table 11 shows that η belongs to h θ , , η i . Shuffle to obtain τ ηη ǫ = h θ , , η i τ ηǫ = θ h , η, τ ηǫ i . Table 11 shows that h , η, τ ηǫ i contains ζ . Finally, θ ζ is detected by τ h Q = h · P h . (cid:3) Lemma . There is no hidden η extension on h p ′ . Proof.
The element h p ′ does not support a hidden τ extension, while Table14 shows that there is a hidden τ extension from τ h C ′′ to ∆ h h c . Therefore,there cannot be a hidden η extension from h p ′ to τ h C ′′ . (cid:3) Lemma . There is a hidden η extension from h d D to τ M d . Proof.
Table 11 shows that h d D detects the Toda bracket h τ κθ . , ν, ν i .Now shuffle to obtain h τ κθ . , ν, ν i η = τ κθ . h ν, ν, η i . Table 11 shows that the Toda bracket h ν, ν, η i contains ǫ . Finally, τ κθ . ǫ isdetected by τ M d because Table 21 shows that there is a hidden ǫ extension from τ M g to M d . (cid:3) Lemma . There is a hidden η extension from h h d to τ d g . Proof.
Table 11 shows that the Toda bracket h η, σ , η, σ i equals κ . Wewould like to consider the shuffle h η, σ , η, σ i τ κ = η h σ , η, σ , τ κ i , but we must show that the Toda bracket h η, σ , τ κ i is well-defined and containszero. It is well-defined because σ κ is detected by h g in π , , and there are no τ extensions on this group. The bracket contains zero by comparison to tmf , sinceall non-zero elements of π , are detected by tmf .We have now shown that τ κ κ is divisible by η . The only possibility is thatthere is a hidden η extension from h h d to τ d g . (cid:3) Lemma . There is no hidden η extension on h x , . Proof.
Table 15 shows that h x , is the target of a hidden 2 extension. (cid:3) Lemma . (1) There is no hidden η extension on h h .(2) There is no hidden η extension on τ m . Proof.
Table 15 shows that e A ′ and τ e A ′ support hidden 2 extensions, sothey cannot be the targets of hidden η extensions. (cid:3) Lemma . There is no hidden η extension on h h d . Proof.
Table 11 shows that h h detects the Toda bracket h θ , , η i , so h h d detects h θ , , η i κ . Now shuffle to obtain h θ , , η i ηκ = θ h , η, ηκ i . Table 11 shows that the Toda bracket h , η, ηκ i equals νκ . Thus we need to computethe product νκθ . Lemma 7.156 shows that this product equals νσ θ , which equalszero. (cid:3) Lemma . There is no hidden η extension on τ h x , . Proof.
Let α be an element of π , that is detected by h x , . Then τ h x , detects τ ηα . Now consider the shuffle τ η α = h , η, i α = 2 h η, , α i . Note that 2 α is zero because there are no 2 extensions in π , , so the secondbracket is well-defined.Finally, 2 h η, , α i must be zero because there are no 2 extensions in π , insufficiently high filtration. (cid:3) Lemma . If h h h supports a hidden η extension, then its target is not τ h x , . Proof.
Table 19 shows that τ h x , supports a hidden ν extension, so itcannot be the target of a hidden η extension. (cid:3) Lemma . If h h h supports a hidden extension, then its target is τ (∆ e + C ) g , and τ h g + τ h e does not survive. .3. HIDDEN η EXTENSIONS 75
Proof.
The element τ h h h is a multiple of h , so it cannot support a hidden η extension. This eliminates all possible targets except for τ (∆ e + C ) g .If τ h g + τ h e survives, then Remark 7.5 shows that τ (∆ e + C ) g supportsa hidden τ extension. As in the previous paragraph, this eliminates τ (∆ e + C ) g as a possible target. (cid:3) Lemma . There is no hidden η extension on h n . Proof.
The element τ h n = h ( τ Q + τ n ) detects σ { τ Q + τ n } . Then ησ { τ Q + τ n } is zero because ησ is zero. (cid:3) Lemma . There is no hidden η extension on ∆ p . Proof.
Table 19 shows that ∆ p is the target of a hidden ν extension, so itcannot be the source of an η extension. (cid:3) Lemma . There is no hidden η extension on h c . Proof.
Table 11 shows that h c detects the Toda bracket h σ, , θ i . By in-spection, all possible indeterminacy is in higher Adams filtration, so h c detectsevery element of the Toda bracket.Shuffle to obtain η h σ, , θ i = h η, σ, i θ . The Toda bracket h η, σ, i is detected in filtration at least 5 since the Masseyproduct h h , c , h i is zero. Therefore, the Toda bracket equals { , ηκ } .We now know that η h σ, , θ i contains zero, and therefore h c does not supporta hidden η extension. (cid:3) Lemma . There is no hidden η extension on h h g . Proof.
Table 11 shows that h h g detects the Toda bracket h ν, η, η κ i . Shuffleto obtain η h ν, η, η κ i = h η, ν, η i η κ. Table 11 shows that h η, ν, η i equals ν . Finally, ν η κ = ν κ h η, , θ i = νθ κ h ν, η, i , which equals zero because h ν, η, i is contained in π , = 0. (cid:3) Lemma . There is no hidden η extension on h h Q . Proof.
We must eliminate τ h gC ′ as a possible target. One might hope touse the homotopy of Cτ in order to do this, but the homotopy of Cτ has an η extension in the relevant degree that could possibly detect a hidden extension from h h Q to τ h gC ′ .If there were a hidden η extension from h h Q to τ h gC ′ , then the hidden τ extension from τ h gC ′ to ∆ h d would imply that there is a hidden η extensionfrom τ h h Q to ∆ h d . However, τ h h Q detects the product ν { τ Q + τ n } ,and ην is zero. Therefore, τ h h Q cannot support a hidden η extension. (cid:3) Lemma . If h x , (resp., h h g , h h g + h f ) does not survive, thenthere is a hidden η extension from τ h x , (resp., τ h h g , τ h h g + τ h f ) toeither τ gQ or to ∆ h d . Proof. If h x , does not survive, then Table 12 shows that τ h x , maps to∆ e + τ ∆ h e g under inclusion of the bottom cell into Cτ . This element supportsan h multiplication to ∆ h d in Cτ . Therefore, τ h x , would also support ahidden η extension.The arguments for h h g and h h g + h f are identical. (cid:3) Lemma . (1) There is no hidden η extension on h h c d .(2) There is no hidden η extension on P h h d . Proof.
Table 15 shows that h h c d and P h h d are targets of hidden 2extensions, so they cannot be the sources of hidden η extensions. (cid:3) Lemma . There is a hidden η extension from h h i to τ ∆ c g . Proof.
The Adams differential d (∆ h ) = ∆ h x implies that τ ∆ c g = h · ∆ h detects the Toda bracket h η, , { ∆ h x }i . However, the later Adamsdifferential d ( h h i ) = ∆ h x implies that 0 belongs to { ∆ h x } . Therefore, τ ∆ c g detects h η, , i , so τ ∆ c g detects a multiple of η . The only possibilityis that there is a hidden η extension from h h i to τ ∆ c g . (cid:3) Lemma . There is no hidden η extension on B d . Proof.
Table 15 shows that B d is the target of a hidden 2 extension, so itcannot be the source of a hidden η extension. (cid:3) Lemma . There is no hidden η extension on h h h c . Proof.
Table 19 shows that h h h c is the target of a hidden ν extension,so it cannot support a hidden η extension. (cid:3) Lemma . There is a hidden η extension from ∆ h f to τ ∆ h c g . Proof.
The element τ ∆ h c g detects the product ν { ∆ t } . Table 11 showsthat ν equals the Toda bracket h η, ν, η i . Shuffle to obtain h η, ν, η i{ ∆ t } = η h ν, η, { ∆ t }i . This shows that τ ∆ h c g is the target of a hidden η extension. The only possiblesource for this extension is ∆ h f . (cid:3) ν extensions Theorem . Table 19 lists some hidden extensions by ν . Proof.
Many of the hidden extensions follow by comparison to Cτ . For ex-ample, there is a hidden ν extension from h g to h c d in the Adams spectralsequence for Cτ . Pulling back along inclusion of the bottom cell into Cτ , theremust also be a hidden ν extension from h g to h c d in the Adams spectral se-quence for the sphere. This type of argument is indicated by the notation Cτ inthe fourth column of Table 17.Next, Table 14 shows a hidden τ extension from h c d to P h d . Therefore,there is also a hidden ν extension from τ h g to P h d . This type of argument isindicated by the notation τ in the fourth column of Table 17. .4. HIDDEN ν EXTENSIONS 77
Some extensions can be resolved by comparison to tmf or to mmf . For example,Table 2 shows that the classical unit map S → tmf takes { ∆ h h } in π to a non-zero element α of π tmf such that να = ηκκ in π tmf . Therefore, there must bea hidden ν extension from ∆ h h to τ h e .The proofs of several of the extensions in Table 19 rely on analogous extensionsin mmf . Extensions in mmf have not been rigorously analyzed [ ]. However, thespecific extensions from mmf that we need are easily deduced from extensions in tmf , together with the multiplicative structure. For example, there is a hidden ν extension in tmf from ∆ h to τ d . Therefore, there is a hidden ν extension in mmf from ∆ h g to τ d g , and also a hidden ν extension from τ ∆ h g to τ d e in thehomotopy groups of the sphere spectrum. Note that mmf really is required here,since d g equals zero in the homotopy of tmf .Many cases require more complicated arguments. In stems up to approxi-mately dimension 62, see [ , Section 4.2.4 and Tables 31–32] and [ ]. Thehigher-dimensional cases are handled in the following lemmas. (cid:3) Remark . The last column of Table 19 indicates which ν extensions arecrossing, as well as which extensions have indeterminacy in the sense of Section2.1.1. Remark . The hidden ν extension from h h d to τ gn is proved in [ ],which relies on the “ R P ∞ -method” to establish a hidden σ extension from τ h d to∆ h c and a hidden η extension from τ h g to ∆ h c . We now have easier proofsfor these η and σ extensions, using the hidden τ extension from h g to ∆ h c given in Table 14, as well as the relation h d = h g . Remark . If M ∆ h d is not hit by a differential, then there is a hidden τ extension from τ M h g from M ∆ h d . This implies that there must be a hidden ν extension from τ (∆ e + C ) g to M ∆ h d . Theorem . Table 20 lists all unknown hidden ν extensions, through the -stem. Proof.
Many possible extensions can be eliminated by comparison to Cτ , to tmf , or to mmf . For example, there cannot be a hidden ν extension from h h h to τ h g by comparison to Cτ .Other possibilities are eliminated by consideration of other parts of the mul-tiplicative structure. For example, there cannot be a hidden ν extension whosetarget supports a multiplication by η , since ην equals zero.Many cases are eliminated by more complicated arguments. These are handledin the following lemmas. (cid:3) Remark . Comparison to synthetic homotopy eliminates several possiblehidden ν extensions, including:(1) from τ h p to τ x , .(2) from ∆ p to τ M h d .See [ ] for more details. Remark . If M ∆ h d is not hit by a differential, then M ∆ h d sup-ports an h extension, and there cannot be a hidden ν extension from h h h h to M ∆ h d . Lemma . There is a hidden ν extension from ∆ e + C to τ M h g . Proof.
Table 11 shows that 2 κ is contained in τ h ν, η, ηκ i . Shuffle to obtainthat ν h η, ηκ, τ θ . i = h ν, η, ηκ i τ θ . , so 2 κθ . is divisible by ν .Table 21 shows that τ M g detects κθ . , so τ M h g detects 2 κθ . . Now we knowthat there is a hidden ν extension whose target is τ M h g , and the only possiblesource is ∆ e + C . (cid:3) Remark . One consequence of the proof of Lemma 7.106 is that ∆ e + C detects the Toda bracket h η, ηκ, τ θ . i . Lemma . There is a hidden ν extension from τ h H to τ M h g . Proof.
Lemma 6.4 shows that the bracket h κ, , η i contains zero with inde-terminacy generated by ηρ . The bracket h τ ηθ . , κ, i equals zero since π , iszero. Therefore, the Toda bracket h τ ηθ . , κ, , η i is well-defined.Table 11 shows that τ g detects h κ, , η, ν i . Therefore, τ M h g detects τ ηθ . h κ, , η, ν i = h τ ηθ . , κ, , η i ν. This shows that τ M h g is the target of a ν extension, and the only possible sourceis τ h H . (cid:3) Remark . The proof of Lemma 7.108 shows that τ h H detects the Todabracket h τ ηθ . , κ, , η i . Lemma . There is no hidden ν extension on h h . Proof.
Table 11 shows that h h detects the Toda bracket h η, , θ i . Shuffleto obtain ν h η, , θ i = h ν, η, i θ = 0 , since h ν, η, i is contained in π , = 0. (cid:3) Lemma . There is no hidden ν extension on h Q . Proof.
Table 17 shows that τ ∆ h e g supports a hidden η extension. There-fore, it cannot be the target of a ν extension. (cid:3) Lemma . There is a hidden ν extension from h A ′ to h h (∆ e + C ) . Proof.
By comparison to Cτ , There cannot be a hidden ν extension from h A ′ to τ h C ′′ + h h (∆ e + C )Table 11 shows that ∆ e + C detects the Toda bracket h η, ηκ, τ θ . i , and h A ′ detects the Toda bracket h ν, η, τ κθ . i . Note that h A ′ also detects h ν, ηκ, τ θ . i .Now shuffle to obtain( ησ + ǫ ) h η, ηκ, τ θ . i + ν h ν, ηκ, τ θ . i = (cid:28)(cid:2) ησ + ǫ ν (cid:3) , (cid:20) ην (cid:21) , ηκ (cid:29) τ θ . . The matric Toda bracket (cid:28)(cid:2) ησ + ǫ ν (cid:3) , (cid:20) ην (cid:21) , ηκ (cid:29) must equal { , ν σ } , since ν σ = { h h c } is the only non-zero element of π , , and that element belongs tothe indeterminacy because it is a multiple of ν . .4. HIDDEN ν EXTENSIONS 79
Next observe that τ ν σθ . is zero because all possible values of σθ . are mul-tiples of η . This shows that ( ησ + ǫ ) α + ν β = 0 , for some α and β detected by ∆ e + C and h A ′ respectively. The product ( ησ + ǫ ) α is detected by h h (∆ e + C ), so there must be a hidden ν extension from h A ′ to h h (∆ e + C ). (cid:3) Lemma . There is no hidden ν extension on h A ′ . Proof.
Table 11 shows that h A ′ detects the Toda bracket h σ, κ, τ ηθ . i . Nowshuffle to obtain ν h σ, κ, τ ηθ . i = h ν, σ, κ i τ ηθ . = h η, ν, σ i τ κθ . . The Toda bracket h η, ν, σ i is zero because it is contained in π , = 0. (cid:3) Lemma . There is no hidden ν extension on p ′ . Proof.
Table 21 shows that p ′ detects the product σθ . Therefore, it cannotsupport a hidden ν extension. (cid:3) Lemma . There is a hidden ν extension from h C ′ to τ d g . Proof.
Let α be an element of π , that is detected by τ X + τ C ′ . Table21 shows that ǫα is detected by d Q , so ηǫα is detected by τ d g . On the otherhand, ησα is zero by comparison to Cτ .Now consider the relation η σ + ν = ηǫ . This shows that ν α is detected by τ d g . Since ν α is detected by τ h C ′ , there must be a hidden ν extension from h C ′ to τ d g . (cid:3) Lemma . There is a hidden ν extension from τ h D ′ to τ M d . Proof.
Table 11 shows that τ h D ′ detects the Toda bracket h η, ν, τ κθ . i .Now shuffle to obtain ν h η, ν, τ κθ . i = h ν, η, ν i τ κθ . . The bracket h ν, η, ν i equals ησ + ǫ [ ].Now we must compute ( ησ + ǫ ) τ κθ . . The product σκ is zero, and Table 21shows that ǫκθ . is detected by M d . These two observations imply that ( ησ + ǫ ) τ κθ . is detected by τ M d . (cid:3) Lemma . There is no hidden ν extension on h c . Proof.
Table 11 shows that h c detects the Toda bracket h ǫ, , θ i . Nowshuffle to obtain ν h ǫ, , θ i = h ν, ǫ, i θ . Finally, the Toda bracket h ν, ǫ, i is zero because it is contained in π , = 0. (cid:3) Lemma . There is a hidden ν extension from h C ′′ to τ g t . Proof.
Let α be an element of π , that is detected by i . Table 19 shows gt detects να . Therefore τ g t detects νκα , so τ g t must be the target of a hidden ν extension. The element h C ′′ is the only possible source for this extension. (cid:3) Lemma . There is no hidden ν extension on M h h g . Proof.
If there were a hidden ν extension from M h h g to τ g t , then therewould also be a hidden ν extension with target τ g t . But there is no possiblesource for such an extension. (cid:3) Lemma . If there is a hidden ν extension on h h d , then its target is M ∆ h h . Proof.
The only other possible target is e A ′ . However, Table 15 shows that e A ′ supports a hidden 2 extension, while h h d does not. (cid:3) Lemma . There is no hidden ν extension on τ d g . Proof.
Table 17 shows that τ d g is the target of a hidden η extension. There-fore, it cannot be the source of a hidden ν extension. (cid:3) Lemma . There is no hidden ν extension on h h A . Proof.
Table 21 shows that h h A detects σ θ , so it cannot support a hidden ν extension. (cid:3) Lemma . There is a hidden ν extension from h h to τ h x . Proof.
Table 11 shows that h h detects h θ , , σ i . Let α be an element of π , that is contained in this Toda bracket. Then να is an element of h θ , , σ i ν = h θ , σ, σ i ν = θ h σ, σ, ν i ⊆ h σ, σθ , ν i . Table 21 shows that p ′ detects σθ . Therefore, the Toda bracket h σ, σθ , ν i isdetected by an element of the Massey product h h h , p ′ , h i . Table 3 shows that h e equals the Massey product h h , p ′ , h i . By inspection of indeterminacy, theMassey product h h h , p ′ , h i contains h e = τ h x with indeterminacy generatedby h h e .We have now shown that να is detected by either τ h x or τ h x + h h e . But h h e = h h d is a multiple of h , so we may add an element in higher Adamsfiltration to α , if necessary, to conclude that να is detected by τ h x . (cid:3) Lemma . There is a hidden ν extension from h h h to τ ∆ h d . Proof.
Table 19 shows that there is a hidden ν extension from h h h to ∆ p .Therefore, there is also a hidden ν extension from h h h to h · ∆ p = τ ∆ h d . (cid:3) Lemma . There is no hidden ν extension on e A ′ . Proof.
A possible hidden ν extension from e A ′ to ∆ h B would be detectedby Cτ , but we have to be careful with the analysis of the homotopy of Cτ becauseof the h extension from ∆ h d g to ∆ h B in the Adams E ∞ -page for Cτ .Let α be an element of π , Cτ that is detected by h C ′ . Then να is detectedby e A ′ , and να maps to zero under projection to the top cell because h C ′ doesnot support a ν extension in the homotopy of the sphere.Therefore, να lies in the image of e A ′ under inclusion of the bottom cell. Since ν α is zero, e A ′ cannot support a hidden ν extension to ∆ h B . (cid:3) Lemma . There is no hidden ν extension on h h h . .4. HIDDEN ν EXTENSIONS 81
Proof.
Table 11 shows that h h h detects the Toda bracket h η , , θ i . Shuf-fle to obtain ν h η , , θ i = h ν, η , i θ . Finally, h ν, η , i must contain zero in π , because tmf detects every element of π , . (cid:3) Lemma . There is no hidden ν extension on h n . Proof.
The element h gD cannot be the target of a hidden ν extension bycomparison to Cτ .The element τ h n = h · h ( τ Q + τ n ) detects a multiple of σ , so it cannotsupport a hidden ν extension. This rules out h gA ′ as a possible target. (cid:3) Lemma . There is no hidden ν extension on τ e g . Proof.
After eliminating other possibilities by comparison to tmf , comparisonto mmf , and by inspection of h multiplications, the only possible target for a hidden ν extension is P h x , .Let α be an element of π , that is detected by e g . Then να is detected by h e g = h h Q . Choose an element β of π , that is detected by h h Q suchthat τ β is zero. Then η β is also detected by h h Q . However, να + η β is notnecessarily zero; it could be detected in Adams filtration at least 13. In any case, τ να equals τ η β = 0 modulo filtration 13. In particular, τ να cannot be detectedby P h x , in filtration 11. (cid:3) Lemma . There is no hidden ν extension on P h h h . Proof.
Table 19 shows that there is a hidden ν extension from P h h to∆ h x . The target of a hidden ν extension on P h h h must have Adams filtra-tion greater than the filtration of ∆ h x . The only possibilities are ruled out bycomparison to tmf . (cid:3) Lemma . There is a hidden ν extension from (∆ e + C ) g to τ M h g . Proof.
Let α be an element of π , that is detected by ∆ e + C . Table 11shows that (∆ e + C ) g detects h α, η , η i . Then ν h α, η , η i = h να, η , η i by inspection of indeterminacies. Table 19 shows that τ M h g detects να . TheToda bracket h να, η , η i is detected by the Massey product h τ M h g, h , h h i = h τ M h g, h , h i = M h g h τ, h , h i = τ M h g . (cid:3) Lemma . There is no hidden ν extension on h c A ′ . Proof.
Table 11 shows that τ h c A ′ detects h τ θ . κ, η, ν i τ σ . Shuffle to obtain h τ θ . κ, η, ν i τ σν = τ θ . κ h η, ν, τ νσ i . The Toda bracket h η, ν, τ νσ i is zero because π , contains only a v -periodicelement detected by P h .We now know that τ h c A ′ does not support a hidden ν extension. In partic-ular, there cannot be a hidden ν extension from τ h c A ′ to M ∆ h e . The hidden τ extension from τ M h g to M ∆ h e implies that there cannot be a hidden ν extension from h c A ′ to τ M h g .Additional cases are ruled out by comparison to Cτ and to mmf . (cid:3) Lemma . (1) There is a hidden ν extension from ∆ j + τ gC ′ to τ M h g .(2) There is a hidden ν extension from τ gC ′ to M ∆ h e . Proof.
Table 19 shows that there exists an element α in π , detected by τ h H such that ν is detected by τ M h g . (Beware that there is a crossing exten-sion here, so not every element detected by τ h H has the desired property.) Table21 shows that τ M h g also detects τ θ . ηκ . However, να does not necessarily equal τ θ . ηκ because the difference could be detected in higher filtration by ∆ h h . Inany case, νκα equals τ θ . ηκ .The product θ . ηκ is detected by τ M h g . The hidden τ extension from τ M h g to M ∆ h e then implies that νκα = τ θ . ηκ is detected by M ∆ h e .We now know that M ∆ h e is the target of a hidden ν extension. The onlypossible source is τ gC ′ . (Lemma 7.131 eliminates another possible source.) Thisestablishes the second extension. The first extension follows from onsideration of τ extensions. (cid:3) Remark . The proof of Lemma 7.132 shows that νκα is detected by M ∆ h e , where α is detected by τ h H . Note that κα is detected by τ h H g = τ h c A ′ . But this does not show that τ h c A ′ supports a hidden ν extension.Rather, it shows that the source of the hidden ν extension is either τ h c A ′ , or anon-zero element in higher filtration. Lemma . There is no hidden ν extension on h h h . Proof.
Table 11 shows that h h h detects the Toda bracket h νν , , θ i .Shuffle to obtain ν h νν , , θ i = h ν, νν , i θ . The Toda bracket h ν, νν , i is zero because π , consists only of a v -periodicelement detected by P h c . (cid:3) Lemma . There is a hidden ν extension from τ h f to h x , + τ g . Proof.
By comparison to Cτ , there must be a hidden ν extension whose targetis either h x , or h x , + τ g .Table 15 shows that there is a hidden 2 extension from τ h f to τ h h Q , andTable 19 shows that there is a hidden ν extension from τ h h Q to τ h g . Thisimplies that the target of the ν extension on τ h f must be h x , + τ g . (cid:3) Lemma . There is no hidden ν extension on x , . Proof.
Let α be an element of π , that is detected by x , . Table 16shows that if 2 α is non-zero, then it is detected in filtration 11. Then 2 να must bedetected in filtration at least 13.The product να cannot be detected by τ g , for then 2 να would be detectedby τ h g in filtration 9. This rules out τ g as a possible target for a hidden ν extension on x , . (cid:3) Lemma . There is no hidden ν extension on P h c . .5. MISCELLANEOUS HIDDEN EXTENSIONS 83 Proof.
Table 21 shows that P h c detects the product ρ η , and νρ η iszero. (cid:3) Lemma . There is a hidden ν extension from h gA ′ to ∆ h g g . Proof.
Comparison to Cτ shows that h gA ′ supports a hidden ν extensionwhose target is either ∆ h g g or ∆ h g g + τ h gC ′′ .Let α be an element of π , that is detected by h gA ′ . Since h gA ′ does notsupport a hidden η extension, we may choose α such that ηα is zero. Note that h gA ′ detects να .Shuffle to obtain ν α = h η, ν, η i α = η h ν, η, α i . This shows that ν α must be divisible by η . Consequently, the hidden ν extensionon h gA ′ must have target ∆ h g g . (cid:3) Remark . The proof of Lemma 7.138 shows that ∆ h g g detects theToda bracket h ν, η, { h gA ′ }i . Lemma . (1) There is a hidden ǫ extension from h h to M c .(2) There is a hidden ǫ extension from τ h h to M P . Proof.
Table 17 shows that
M h detects the product ηθ . . Then M h c detects ηǫθ . . This implies that M c detects ǫθ . .This only shows that M c is the target of a hidden ǫ extension, whose sourcecould be h h or h d . However, Lemma 7.147 rules out the latter case. Thisestablishes the first hidden extension.Table 14 shows that there is a hidden τ extension from M c to M P . Then thefirst hidden extension implies the second one. (cid:3)
Remark . We claimed in [ , Table 33] that there is a hidden ǫ extensionfrom h h to M c . However, the argument given in [ , Lemma 4.108] only impliesthat M c is the target of a hidden extension from either h h or h d . Lemma . There is a hidden κ extension from h h to M d . Proof.
Table 17 shows that
M h detects the product ηθ . . Then M h d detects the product ηκθ . , so M d must detect the product κθ . . This shows that M d is the target of a hidden κ extension whose source is either h h or h d .We showed in Lemma 7.147 that ǫα is zero for some element α of π , thatis detected by h d . Then ǫκα is also zero. Table 21 shows that ǫκ equals κ .Therefore, κ α is zero. If κα were detected by M d , then κ α would be detectedby M d . It follows that there is no hidden κ extension from h d to M d . (cid:3) Remark . We showed in [ , Table 33] that there is a hidden κ extensionfrom either h h or h d to M d . Lemma 7.142 settles this uncertainty. Lemma . There is a hidden κ extension from h h to τ M g . Proof.
Table 17 shows that
M h detects the product ηθ . . Then τ M h g detects the product ηκθ . , so τ M g must detect the product κθ . . This shows that τ M g is the target of a hidden κ extension whose source is either h h or h d .We showed in Lemma 7.147 that ǫα is zero for some element α of π , thatis detected by h d . If κα were detected by τ M g , then ǫκα would be detected by M d because Table 21 shows that there is a hidden ǫ extension from τ M g to M d .Therefore, there is no hidden κ extension from h d to τ M g . (cid:3) Lemma . There is a hidden { ∆ h h } extension from h h to M ∆ h h . Proof.
Table 17 shows that
M h detects the product ηθ . . Therefore, theelement M ∆ h h detects η { ∆ h h } θ . . This shows that M ∆ h h is the targetof a hidden { ∆ h h } extension. Lemma 7.148 rules out h d as a possible source.The only remaining possible source is h h . (cid:3) Lemma . There is a hidden θ . extension from h h to M . Proof.
The proof of Lemma 5.28 shows that M h detects a multiple of ηθ . .Therefore, it detects either ηθ . or ηθ . { h d } .Now h h d detects η { h d } , which also detects η θ by Table 21. In fact,the proof of [ , Lemma 4.112] shows that these two products are equal. Then ηθ . { h d } equals η θ θ . . Next, η θ . lies in π , . The only non-zero elementof π , is detected by mmf , so the product η θ . must be zero.We have now shown that M h detects ηθ . . This implies that M detects θ . . (cid:3) Lemma . There is no hidden ǫ extension on h d . Proof.
Table 11 shows that h d detects the Toda bracket h , θ , κ i . Nowshuffle to obtain ǫ h , θ , κ i = h ǫ, , θ i κ. Table 11 shows that h c detects the Toda bracket h ǫ, , θ i , and there is no inde-terminacy. Let α in π , be the unique element of this Toda bracket. We wish tocompute ακ .Table 11 shows that h c also detects the Toda bracket h η , ν, ν i , with inde-terminacy generated by ση . Let β in π , be an element of this Toda bracket.Then α and β are equal, modulo ση and modulo elements in higher filtration.Both τ h d and τ c g detect multiples of σ . Also, the difference between α and β cannot be detected by ∆ h d by comparison to tmf .This implies that α equals β + σγ for some element γ in π , . Then ακ = ( β + σγ ) κ = βκ because σκ is zero.Now shuffle to obtain βκ = h η , ν, ν i κ = η h ν, ν, κ i . Table 11 shows that h ν, ν, κ i contains ηκ , and its indeterminacy is generated by νν . We now need to compute η ηκ .The product η κ is detected by τ h h g = τ h h g , so η κ equals τ ησκ , moduloelements of higher filtration. But these elements of higher filtration are eitherannihilated by η or detected by tmf , so η ηκ equals τ η σκ . By comparison to tmf ,this latter expression must be zero. (cid:3) .5. MISCELLANEOUS HIDDEN EXTENSIONS 85 Lemma . There is no hidden { ∆ h h } extension on h d . Proof.
Table 11 shows that h d detects the Toda bracket h κ, θ , i . Byinspection of indeterminacies, we have { ∆ h h }h κ, θ , i = h{ ∆ h h } κ, θ , i . Table 21 shows that τ d l + ∆ c d detects the product { ∆ h h } κ . Now apply theMoss Convergence Theorem 2.16 with the Adams differential d ( h ) = h h todetermine that the Toda bracket h{ ∆ h h } κ, θ , i is detected in Adams filtrationat least 13.The only element in sufficiently high filtration is τ e g , but comparison to mmf rules this out. Thus the Toda bracket h{ ∆ h h } κ, θ , i contains zero. (cid:3) Lemma . There is a hidden κ extension from h to h h A . Proof.
Table 21 shows that there is a hidden σ extension from h to h h A .Lemma 7.156 implies that there also must be a hidden κ extension from h to h h A . (cid:3) Lemma . There is a hidden ρ extension from h to either h x , or τ m . Proof.
Table 11 shows that the Toda bracket h , σ, σ i contains ρ . Then ρ is also contained in h , σ, σ i , although the indeterminacy increases.Now shuffle to obtain ρ θ = θ h , σ, σ i = h θ , , σ i σ. Table 11 shows that h h h detects h θ , , σ i . Also, there is a σ extension from h h h to h x , in the homotopy of Cτ .This implies that ρ θ is non-zero in π , , and that it is detected in filtrationat most 8. Moreover, it is detected in filtration at least 7, since ρ and θ aredetected in filtrations 4 and 2 respectively.There are several elements in filtration 7 that could detect ρ θ . The element x , (if it survives to the E ∞ -page) is ruled out by comparison to Cτ . The element τ h x , is ruled out because ηρ θ is detected in filtration at least 10, since ηρ is detected in filtration 7.The only remaining possibilities are h x , and τ m . (cid:3) Lemma . (1) There is a hidden ρ extension from h h to P h c .(2) There is a hidden ρ extension from h h to P h c . Proof.
Table 11 shows that h h detects h η, , θ i . Then ρ h η, , θ i ⊆ h ηρ , , θ i . The last bracket is detected by
P h c because d ( h ) = h h and because P c detects ηρ . Also, its indeterminacy is in Adams filtration greater than 8. Thisestablishes the first hidden extension.The proof for the second extension is essentially the same, using that P c detects ηρ and that the indeterminacy of h ηρ , , θ i is in Adams filtration greaterthan 12. (cid:3) Lemma . There is a hidden ǫ extension from τ M g to M d . Proof.
First, we have the relation c · h X = M h h g in the Adams E -page,which is detected in the homotopy of Cτ . Table 14 shows that there are hidden τ extensions from h X and M h h g to τ M g and M d respectively. (cid:3) Lemma . If M ∆ h d is non-zero in the E ∞ -page, then there is a hidden ǫ extension from M ∆ h h to M ∆ h d . Proof.
Table 17 shows that
M h detects the product ηθ . . Since M ∆ h d equals ∆ h d · M h , it detects η { ∆ h d } θ . .Table 21 shows that η { ∆ h d } equals ǫ { ∆ h h } , since they are both detectedby ∆ h d and there are no elements in higher Adams filtration. Therefore, theproduct ǫ { ∆ h h } θ . is detected by M ∆ h d . In particular, { ∆ h h } θ . is non-zero, and it can only be detected by M ∆ h h . (cid:3) Lemma . The product ( ησ + ǫ ) θ is detected by τ h h Q . Proof.
Table 11 indicates a hidden η extension from h h to τ h h Q . There-fore, there exists an element α in π , such that τ h h Q detects ηα . (Bewareof the crossing extension from p ′ to h p ′ . This means that it is possible to choosesuch an α , but not any element detected by h h will suffice.)Table 11 shows that h h detects the Toda bracket h ν , , θ i . Let β be an ele-ment of this Toda bracket. Since α and β are both detected by h h , the difference α − β is detected in Adams filtration at least 4.Table 21 shows that p ′ detects σθ , which belongs to the indeterminacy of h ν , , θ i . Therefore, we may choose β such that the difference α − β is detectedin filtration at least 9. Since ηα is detected by τ h h Q in filtration 7, it followsthat ηβ is also detected by τ h h Q .We now have an element β contained in h ν , , θ i such that ηβ is detected by τ h h Q . Now consider the shuffle η h ν , , θ i = h η, ν , i θ . Table 11 shows that the last bracket equals { ǫ, ǫ + ησ } . Therefore, either ǫθ or( ǫ + ησ ) θ is detected by τ h h Q . But ǫθ is detected by h p ′ = h c . (cid:3) Lemma . There exists an element α of π , that is detected by h Q + h n such that τ να equals ( ησ + ǫ ) θ . Proof.
Lemma 7.154 shows that τ h h Q detects ( ǫ + ησ ) θ . The element τ h h Q also detects τ να . Let β be the difference τ να − ( ǫ + ησ ) θ , which isdetected in higher Adams filtration. We will show that β must equal zero.First, τ h D ′ cannot detect β because η β is zero, while Table 17 shows that τ d g detects η { τ h D ′ } . Second, Table 19 shows that τ h h (∆ e + C ) is thetarget of a hidden ν extension. Therefore, we may alter the choice of α to ensurethat β is not detected by τ h h (∆ e + C ). Third, ∆ h c is also the target ofa ν extension. Therefore, we may alter the choice of α to ensure that β is notdetected by ∆ h c . Finally, comparison to tmf implies that β is not detected by τ ∆ h g + τ ∆ h g . (cid:3) Lemma . The product ( σ + κ ) θ is zero. .6. ADDITIONAL RELATIONS 87 Proof.
Table 11 shows that h detects the Toda bracket h , θ , θ , i . Shuffleto obtain h , θ , θ , i ( σ + κ ) = 2 h θ , θ , , σ + κ i . The second bracket is well-defined because h θ , , σ + κ i contains zero [ , ?]. Next, h θ , θ , , σ + κ i h θ , θ , α i for some α in h , σ + κ, ii .The Toda bracket h , σ + κ, i equals h , σ , i + h , κ, i , which equals τ ηκ by Corollary 6.2 since ησ equals zero. The indeterminacy isgenerated by 2 ρ . Thus, α equals τ ηκ + 2 kρ for some integer k . In every case,the bracket h θ , θ , α i has no indeterminacy, so it equals h θ , θ , κ i τ η + h θ , θ , i kρ . The second term is zero since h θ , θ , i is contained in π , = 0. The first bracket h θ , θ , κ i is zero by Lemma 6.20. (cid:3) Lemma . ησ { k } + ν { d e } = 0 in π , . Proof.
We have the relation h h k + h d e = 0 in the Adams E ∞ -page, but ησ { k } + ν { d e } could possibly be detected in higher Adams filtration. However,it cannot be detected by h C ′′ or M h h g by comparison to Cτ . Also, it cannotbe detected by ∆ h d e by comparison to mmf . (cid:3) HAPTER 8
Tables
Table 1 gives some notation for elements in π ∗ , ∗ . The fourth column givespartial information that reduces the indeterminacies in the definitions, but doesnot completely specify a unique element in all cases. See Section 1.5 for furtherdiscussion.Table 2 gives hidden values of the unit map π ∗ , ∗ → π ∗ , ∗ mmf . The elements inthe third column belong to the Adams E ∞ -page for mmf [ ] [ ]. See Section 2.2for further discussion.Table 3 lists information about some Massey products. The fifth column in-dicates the proof. When a differential appears in this column, it indicates theMay differential that can be used with the May Convergence Theorem (see Remark2.26) to compute the bracket. The sixth column shows where each specific Masseyproduct is used in the manuscript. See Chapter 4 for more discussion.Table 4 lists all of the multiplicative generators of the Adams E -page throughthe 95-stem. The third column indicates the value of the d differential, if it isnon-zero. A blank entry in the third column indicates that the d differential iszero. The fourth column indicates the proof. A blank entry in the fourth columnindicates that there are no possible values for the differential. The fifth columngives alternative names for the element, as used in [ ], [ ], or [ ]. See Sections1.5 and 5.1 for further discussion.Table 5 lists some elements in the Adams spectral sequence that are known tobe permanent cycles. The third column indicates the proof. When a Toda bracketappears in the third column, the Moss Convergence Theorem 2.16 applied to thatToda bracket implies that the element is a permanent cycle (see Table 11 for moreinformation). When a product appears in the third column, the element mustsurvive to detect that product.Table 6 lists the multiplicative generators of the Adams E -page through the95-stem whose d differentials are non-zero, or whose d differentials are zero fornon-obvious reasons. See Section 5.2 for further discussion.Table 7 lists the multiplicative generators of the Adams E -page through the95-stem whose d differentials are non-zero, or whose d differentials are zero fornon-obvious reasons. See Section 5.3 for further discussion.Table 8 lists the multiplicative generators of the Adams E -page through the95-stem whose d differentials are non-zero, or whose d differentials are zero fornon-obvious reasons. See Section 5.4 for further discussion.Table 9 lists the multiplicative generators of the Adams E r -page, for r ≥ d r differentials are non-zero, or whose d r differentialsare zero for non-obvious reasons. See Section 5.5 for further discussion.Table 10 lists the possible Adams differentials that remain unresolved. Thesepossibilities also appear in Tables 4–9.
890 8. TABLES
Table 11 lists information about some Toda brackets. The third column ofTable 11 gives an element of the Adams E ∞ -page that detects an element of theToda bracket. The fourth column of Table 11 gives partial information aboutindeterminacies, again by giving detecting elements of the Adams E ∞ -page. Wehave not completely analyzed the indeterminacies of some brackets when the detailsare inconsequential for our purposes; this is indicated by a blank entry in the fourthcolumn. The fifth column indicates the proof of the Toda bracket, and the sixthcolumn shows where each specific Toda bracket is used in the manuscript. SeeChapter 6 for further discussion.Tables 12 and 13 gives hidden values of the inclusion π ∗ , ∗ → π ∗ , ∗ Cτ of thebottom cell, and of the projection π ∗ , ∗ Cτ → π ∗− , ∗ +1 to the top cell. See Section7.1 for further discussion.Table 14 lists hidden τ extensions in the E ∞ -page of the C -motivic Adamsspectral sequence. See Section 7.1 for further discussion.Tables 15, 17, and 19 list hidden extensions by 2, η , and ν . The fourth columnindicates the proof of each extension. The fifth column gives additional informationabout each extension, including whether it is a crossing extension and whether ithas indeterminacy in the sense of Section 2.1.1. See Sections 7.2, 7.3, and 7.4 forfurther discussion.Tables 16, 18, and 20 list possible hidden extensions by 2, η , and ν that wehave not yet resolved.Finally, Table 21 gives some various hidden extensions by elements other than2, η , and ν . See Section 7.5 for further discussion. . TABLES 91 Table 1: Notation for π ∗ , ∗ ( s, w ) element detected by definition(0 , − τ τ (0 ,
0) 2 h (1 , η h (3 , ν h (7 , σ h (8 , ǫ c (9 , µ P h (14 , κ d (15 , ρ h h (16 , η h h (17 , µ P h (19 , σ c (20 , κ τ g h κ, , η, ν i (23 , ρ h i + τ P h d (25 , µ P h (30 , θ h (32 , η h h in h η, , θ i (32 , κ d (44 , κ g (45 , θ . h ηθ . ∈ { M h } (62 , θ h (63 , η h h in h η, , θ i Table 2: Some hidden values of the unit map of mmf ( s, f, w ) element image(28 , , h h g cg (29 , , h h g h cg (32 , ,
17) ∆ h h ∆ c + τ ag (33 , ,
18) ∆ h h ∆ h c (35 , , τ h e P an (40 , , τ ∆ h d P (∆ c + τ ag )(48 , , h h g cg (49 , , h h g h cg (52 , ,
29) ∆ h h g (∆ c + τ ag ) g (53 , ,
30) ∆ h h g ∆ h cg (54 , , h h i ∆ h (54 , , h h e dg (55 , , h h e h dg (57 , , h h h i ∆ h (∆ c + τ ag )(59 , , P h h e ∆ h dg (60 , , τ h g ∆ h dg (62 , , h h c e cdg (63 , , h h c e h cdg (65 , , P h j ∆ h d (66 , , P h h c e (∆ c + τ ag ) dg (67 , , P h h c e ∆ h cdg (68 , , P h h j ∆ h d (68 , , h h g cg (69 , , h h g h cg (71 , ,
38) ∆ h h g ∆ h (∆ c + τ ag ) d (72 , ,
41) ∆ h h g (∆ c + τ ag ) g (73 , ,
42) ∆ h h g ∆ h cg (77 , ,
42) ∆ h g ∆ h (∆ c + τ ag ) g (88 , , h h g cg (89 , , h h g h cg . T A B L E S Table 3: Some Massey products in Ext C ( s, f, w ) bracket contains indeterminacy proof used for(2 , , h h , h , h i τ h h , η, i (3 , , h h , h , h i h h h{ h x , } , , η i , 7.71(6 , , h h , h , h i h h η, ν, η i (8 , , h h , h , h i h h h ν, η, ν i (8 , , h h , h , h , h i c d ( h ) = h h h η , , η, ν i d ( h ) = h h (8 , , h h , h , h h i c d ( h (1) = h h h η, ν, ν i , h η , ν, ν i (9 , , h h , h , h h i P h d ( b ) = h h h η, , σ i (11 , , h h , h , τ h c i P h d ( b h (1)) = τ h c h , η, τ ηǫ i (20 , , h τ, h , h i τ g d ( g ) = h h , , h h , h , h h i h i + τ P h d d ( b ) = h h h σ, , ρ i (30 , , h h , h , h , h i ∆ h d ( h b ) = h h , , h h , h , h , h i d d ( h (1)) = h h h η, σ , η, σ i (33 , , h h h , h , h i p d ( h b ) = h h h ηθ , η, i (46 , , h h , h , h g i M h M , , h h , h , D i τ ∆ h ] Example 2.31(66 , , h h , h , h , h i ∆ h , , h P d , h , g i M P d M , , h h g, h , g i M h g M , , h h h , h , τ gn i τ g n M h h g Lemma 4.8 5.40(75 , , h ∆ h d e , h , h h i ∆ h d e h τ ηκκ , η, η η i (80 , , h h , p ′ , h i h e , , h ∆ e + C , h , h h i (∆ e + C ) g h{ ∆ e + C } , η , η i (86 , , h ∆ h e , h , h g i M ∆ h e M , , h M h , h , h g i M h M , , h τ gG , h h , h i ? τ e x , τ M h Lemma 4.12 5.60
Table 4: E -page generators of the C -motivic Adams spectral se-quence( s, f, w ) element d proof other names(0 , , h (1 , , h (3 , , h (7 , , h (8 , , c (9 , , P h (11 , , P h (14 , , d (15 , , h h h Cτ (16 , , P c (17 , , e h d Cτ (17 , , P h (18 , , f h e Cτ (19 , , c Cτ (19 , , P h (20 , , τ g (22 , , P d (23 , , h g (23 , , i P h d Cτ (24 , , P c (25 , , P e P h d Cτ (25 , , P h (26 , , j P h e Cτ (27 , , h g h h g Cτ (27 , , P h (29 , , k h d Cτ (30 , ,
16) ∆ h Cτ r (30 , , P d (31 , , h h h Cτ (31 , , n (32 , , d (32 , ,
17) ∆ h h h q (32 , , l h d e Cτ (32 , , P c (33 , , p (33 , , P e P h d Cτ (33 , , P h (34 , , P j P h e Cτ (35 , , m h e Cτ (35 , , P h (36 , , t Cτ (37 , , x (37 , , e g h e Cτ . TABLES 95 Table 4: E -page generators of the C -motivic Adams spectral se-quence( s, f, w ) element d proof other names(38 , , e (38 , ,
20) ∆ h h x Cτ y +? h d (38 , , P d (39 , , c g Cτ (39 , ,
21) ∆ h d u (39 , , P i P h d Cτ (40 , , f (40 , , τ g (40 , , P c (41 , , c h f Cτ (41 , ,
22) ∆ h e Cτ z (41 , , P e P h d Cτ (41 , , P h (42 , ,
23) ∆ h e ∆ h d Cτ v (42 , , P j P h e Cτ (43 , , h g (43 , , P h (44 , , g (45 , , τ ∆ h g w (46 , , M h B (46 , ,
25) ∆ h c N (46 , ,
25) ∆ c d τ h d e Cτ u ′ (46 , , P d (47 , , h g h h g Cτ (47 , ,
24) ∆ h i h i Cτ Q ′ , Q + P u (47 , , P ∆ h d (48 , , M h B +? h h e (48 , , P c (49 , ,
27) ∆ c e ∆ h c d + τ h d e Cτ v ′ (49 , , P e P h d Cτ (49 , , P h (50 , , C (50 , ,
28) ∆ h g Cτ (50 , , P ∆ h e P ∆ h d Cτ (50 , , P j P h e Cτ (51 , ,
28) ∆ h g ∆ h h g Cτ G (51 , , gn (51 , , P h (52 , , D h h g Cτ (52 , , d g (53 , , i Cτ (53 , , M c h B , P h d (53 , , M P x ′ (54 , , τ ∆ h M h h Cτ G
Table 4: E -page generators of the C -motivic Adams spectral se-quence( s, f, w ) element d proof other names(54 , ,
28) ∆ h M P h Cτ R +? h h i (54 , , P ∆ c d τ P h d e Cτ (54 , , P d (55 , , B Cτ (55 , , gm h e g Cτ (55 , , P ∆ h d (55 , , P i P h d Cτ (56 , ,
29) ∆ h h τ M P h h Q +? gt (56 , , gt Cτ (56 , , P c (57 , , D h B Cτ (57 , , Q Cτ (57 , ,
31) ∆ h d D (57 , , P ∆ c e P ∆ h c d + τ h d Cτ (57 , , P e P h d Cτ (57 , , P h (58 , , D h Q Cτ (58 , , e g (58 , , P ∆ h e P ∆ h d Cτ (58 , , P j P h e Cτ (59 , , j (59 , , M d B (59 , , c g (59 , , P h (60 , , M h Cτ B (60 , , B M h d Cτ (60 , , τ g (60 , , h g (61 , , D (61 , , A ′ Cτ (61 , , A + A ′ M h h Cτ (61 , , B (61 , ,
32) ∆ x h B + τ M h d Lemma 5.3 X or Cτ , h (62 , , H B Cτ (62 , , C x , + h h (62 , ,
33) ∆ e E , x , + x , (62 , ,
32) ∆ h Cτ x , + x , ++ h X , R (62 , , M e M h d Cτ B , x , (62 , , P ∆ c d τ P h d e Cτ (62 , , P d (63 , , h h h Cτ (63 , , C ′ Cτ x , + x ,
44. TABLES 97
Table 4: E -page generators of the C -motivic Adams spectral se-quence( s, f, w ) element d proof other names(63 , , X M h h Cτ x , (63 , , h g (63 , , P ∆ h d (64 , , A ′′ h X Cτ (64 , ,
33) ∆ h h x , +? h h Q , q (64 , ,
34) ∆ h d M P h Cτ U , P Q +? km (64 , , P c (65 , , k (65 , , τ M g Cτ B (65 , ,
34) ∆ h e h · ∆ h d Cτ R +? gw (65 , , P ∆ c e P ∆ h c d + Cτ + τ P h d (65 , , P e P h d Cτ (65 , , P h (66 , ,
36) ∆ h r (66 , , τ G h C + h h Q Cτ x , +? h r (66 , , D ′ τ M h g Cτ , i P D (66 , , τ B τ M h g i (66 , , P ∆ h e P ∆ h d Cτ (66 , , P j P h e Cτ (67 , , τ Q τ ∆ h h g (67 , , n ∆ h h Cτ (67 , , h Q + h D Cτ (67 , , X Cτ x , (67 , , C ′′ x , (67 , ,
35) ∆ c x , + h x , (67 , , h g h h g Cτ (67 , , P h (68 , , d Cτ (68 , ,
36) ∆ g h X Cτ G +? h h A ′ (68 , , M h g (68 , ,
36) ∆ h g M P h d Cτ P D +? h G , G (69 , , p ′ Cτ (69 , , D ′ h X +? τ h C ′′ Cτ P D (69 , , h G h C ′′ Cτ (69 , , P ( A + A ′ ) τ M h h g Cτ , i (69 , , h B Cτ (69 , , τ ∆ h g d W (69 , , M P x , +? d il (70 , , p Cτ (70 , , h Q Cτ (70 , , P ∆ h d M P h Cτ R ′ , R +? d v Table 4: E -page generators of the C -motivic Adams spectral se-quence( s, f, w ) element d proof other names(70 , , P ∆ c d τ P h d e Cτ (70 , , P d (71 , , x , τ d e h x , +? h p ′ (71 , , l (71 , ,
37) ∆ h c Cτ , tmf , h x , +? h d Q (71 , ,
38) ∆ h g Cτ x , (71 , , M j M P h e Cτ x , (71 , ,
40) ∆ h g h m Cτ (71 , , g n (71 , , P ∆ h d (71 , , P i P h d Cτ (72 , , d g (72 , ,
39) ∆ h c d M P h c Cτ (72 , , P ∆ h d M P h Cτ (72 , , P c (73 , , P ∆ h e M P h Cτ (73 , , P ∆ c e P ∆ h c d + Cτ + τ P h d (73 , , P e P h d Cτ (73 , , P h (74 , , x , x , +? P h h h (74 , , P ∆ h e P ∆ h d Cτ (74 , , P j P h e Cτ (75 , , x , Cτ x , (75 , , gB Cτ (75 , ,
40) ∆ h g ∆ h h g Cτ (75 , , g m h e g Cτ (75 , , P h (76 , , x , h x , Cτ x , (76 , , x , Cτ x , +? h h B (76 , , g t Cτ (76 , ,
40) ∆ d τ d jm Cτ , tmf x , (77 , , x , τ M h h Lemma 5.5 x , + m (77 , , m Cτ (77 , , x , ∆ h h g Cτ x , (77 , , M ∆ h h Cτ , h , τ g P D (77 , ,
43) ∆ h d g (77 , ,
40) ∆ h k ∆ h d Cτ x , +? e g (77 , , e g h e g Cτ (78 , , t h m Cτ (78 , , x , Cτ x , +? h h h (78 , , x , h x , Cτ , h P h (78 , , e g (78 , , P ∆ c d τ P h d e Cτ . TABLES 99 Table 4: E -page generators of the C -motivic Adams spectral se-quence( s, f, w ) element d proof other names(78 , , P d (79 , , x Cτ (79 , ,
42) ∆ B Cτ (79 , , gj (79 , ,
41) ∆ n x , (79 , , c g Cτ (79 , ,
42) ∆ d e ∆ h d + τ d km Cτ , tmf x , (79 , , P ∆ h i P h i Cτ (79 , , P ∆ h d Cτ (80 , , e h x Cτ (80 , ,
42) ∆ d h x , (80 , , gB M h e Cτ (80 , ,
41) ∆ h h h x , (80 , ,
42) ∆ h l ∆ h d e Cτ x , +? g (80 , , τ g (80 , , P ∆ h d M P h Cτ (80 , , P c (81 , , gA ′ (81 , , gB (81 , ,
42) ∆ p h x , (81 , , P ∆ h e h · P ∆ h d Cτ (81 , , P ∆ c e P ∆ h c d + Cτ + τ P h d (81 , , P e P h d Cτ (81 , , P h (82 , , h Q (82 , , gH gB Cτ (82 , ,
45) (∆ e + C ) g (82 , , gC (82 , , M e g M h e Cτ (82 , ,
44) ∆ e τ ∆ h e Cτ , tmf x , (82 , ,
47) ∆ h e g ∆ h e g + M h d e Cτ (82 , , P ∆ h e P ∆ h d Cτ (82 , , P j P h e Cτ (83 , ,
45) ∆ j (83 , , gC ′ (83 , ,
44) ∆ m ∆ h e + Cτ , tmf x , + τ ∆ h e g (83 , , h g (83 , , P h (84 , , f Cτ (84 , , P x , Cτ (84 , ,
44) ∆ t Cτ x , (84 , ,
44) ∆ h B ∆ h m + τ ∆ h e Cτ , mmf x , + x , Table 4: E -page generators of the C -motivic Adams spectral se-quence( s, f, w ) element d proof other names(84 , , M ∆ h d Cτ x , (85 , , c h f Cτ (85 , , x , h x , + h c (85 , ,
44) ∆ x Cτ , h x , (85 , , M g Cτ (85 , ,
46) ∆ e g ∆ h e + τ d m tmf x , + h x , (85 , , M P (86 , , τ gG h (∆ e + C ) g Cτ (86 , ,
45) ∆ e d x , (86 , ,
46) ∆ h B (86 , ,
44) ∆ h ∆ h x Cτ , h P h +? gB (86 , , τ B g τ M h g Lemma 5.6(86 , ,
44) ∆ P h d M P h Cτ x , +? P d v (86 , , P ∆ c d τ P h d e h (86 , , P d (87 , , x , x , (87 , , gQ Cτ (87 , ,
46) ∆ h H ∆ h B Cτ x , (87 , , gC ′′ Cτ (87 , ,
47) ∆ c g Cτ (87 , , M ∆ h e M ∆ h d Cτ x , +? h x , (87 , ,
45) ∆ h d τ e m tmf x , (87 , , h g h h g Cτ (87 , , P ∆ d e P ∆ h d + Cτ , tmf + τ ∆ h d (87 , , P ∆ h d Cτ (87 , , P i P h d Cτ (88 , , x , (88 , ,
46) ∆ f x , +? gG ++? P h h e (88 , ,
48) ∆ g g (88 , , τ ∆ g τ ∆ h e g tmf , h x , (88 , , P ∆ h d M P h Cτ (88 , , P c (89 , ,
46) ∆ c ∆ h f Cτ x , (89 , , h gG h gC ′′ Cτ (89 , , h B g Cτ (89 , ,
46) ∆ h e τ e g Cτ , tmf x , (89 , ,
51) ∆ c e g ∆ h c e g + τ h e g Cτ , h + M h c d e (89 , , P ∆ h e M P h Cτ (89 , , P ∆ c e P ∆ h c d + Cτ + τ P h d (89 , , P e P h d Cτ . TABLES 101 Table 4: E -page generators of the C -motivic Adams spectral se-quence( s, f, w ) element d proof other names(89 , , P h (90 , , x , x , (90 , , M x , (90 , , M ∆ h g Cτ rB (90 , ,
47) ∆ h e ∆ h d + τ e gm Cτ , tmf x , (90 , , P ∆ h e P ∆ h d Cτ (90 , , P j P h e Cτ (91 , , x , x , Cτ x , (91 , , x , M h Cτ x , +? h h d (91 , , M ∆ c d τ M h d e Cτ , h (91 , ,
50) ∆ h g (91 , ,
52) ∆ h g h gm Cτ (91 , , g n (91 , , P h (92 , , g Cτ (92 , , x , Cτ x , +? h h k (92 , ,
51) ∆ h e Cτ (92 , ,
48) ∆ g Cτ x , +? h x , (92 , , d g (92 , ,
48) ∆ h d τ d e g Cτ , tmf x , (92 , , P ∆ d τ d ij Cτ , tmf (93 , , x , Cτ x , + h h r (93 , ,
51) ∆ h e (93 , ,
49) ∆ h H h x , Cτ x , + h h r (93 , ,
50) ∆∆ e M h Cτ x , +? P h c d (93 , , g i Cτ (93 , , τ ∆ h g τ e gm tmf , h x , (94 , , x , ? τ h x , x , (94 , , x , Cτ (94 , , y , Cτ x , +? h x , (94 , , x , (94 , ,
49) ∆ M h x , (94 , ,
49) ∆ h c d x , (94 , , M ∆ c e M ∆ h c d + Cτ , h + τ M h d e (94 , ,
49) ∆ c d τ ∆ h d e h (94 , ,
50) ∆ d l ∆ h d e + Cτ , tmf x , + τ ∆ h e (94 , ,
48) ∆ i τ d e tmf x , (94 , , P ∆ c d τ P h d e Cτ (94 , , P d (95 , , x , x , +? τ d H Cτ x , (95 , , x , x , Cτ (95 , ,
50) ∆ x , τ ∆ d e Cτ , h x , +? h h l
02 8. TABLES
Table 4: E -page generators of the C -motivic Adams spectral se-quence( s, f, w ) element d proof other names(95 , , x , Cτ (95 , , g B (95 , , M ∆ h g Cτ (95 , ,
52) ∆ h g P h h c Cτ (95 , ,
50) ∆ h e τ e g Cτ , tmf x , (95 , , g m h e g Cτ (95 , ,
48) ∆ h i ∆ h i + τ d e m Cτ , tmf x , +? P x , (95 , , P ∆ h d τ d e m tmf (95 , , P ∆ d e P ∆ h d + τ d ik Cτ , tmf (95 , , P ∆ h d Cτ . TABLES 103 Table 5: Some permanent cycles in the C -motivic Adams spectralsequence( s, f, w ) element proof(36 , , t h τ, η κ , η i (64 , , h h h η, , θ i (68 , , h A ′ h σ, κ, τ ηθ . i (69 , , h h h ν , , θ i (69 , , p ′ σθ (70 , , h h h h σ, , θ i (70 , , τ ∆ h g + τ ∆ h g h η, τ κ , τ κ i (71 , , h c h ǫ, , θ i (72 , , P h h h µ , , θ i (74 , , P h h h Lemma 5.68(74 , , x , θ κ (77 , , h h h σ , , θ i (77 , , h d h κ, , θ i (79 , , h h h h η , , θ i (79 , , P h c ρ η (80 , , τ h x h , η, τ η { h x , }i (80 , , P h h h µ , , θ i (80 , ,
42) ∆ d Lemma 5.19(82 , , h c h σ, , θ i (83 , , h h g h κη , η, ν i (84 , , h h h h νν , , θ i (85 , , τ h f Lemma 5.46(85 , , P h d h τ η κ, , θ i (86 , , h h c σ { h h h } (87 , , h c h τ { h Q + h n } , ν , η i (87 , , P h c ρ η (88 , , x , Lemma 5.73(88 , , P h h h µ , , θ i (90 , , M Lemma 5.28(91 , , h h h g h{ h h g } , , θ i (92 , , h g Lemma 5.76(93 , , h h h h θ , , θ i (93 , , h x , h κ , κ, τ ηθ . i (94 , , h n h{ n } , , θ i (95 , , h d h κ , , θ i (95 , ,
49) ∆ h h h h{ ∆ h h } , , θ i (95 , , P h c Lemma 5.32
04 8. TABLES
Table 6: Adams d differentials in the C -motivic Adams spectralsequence( s, f, w ) element d proof(15 , , h h h d Cτ (30 , ,
16) ∆ h τ h d tmf (31 , , h h h · ∆ h Cτ (31 , , τ d e P c d d ( τ d e + h h )(34 , , h h τ h d Lemma 5.12(37 , , τ e g c d d ( τ e g )(38 , , e h t Cτ (39 , , τ P d e P c d h (40 , , f h (46 , , i τ P h d tmf (47 , , τ P d e P c d h (47 , , h Q ′ P h d Cτ (49 , , h h e M h Cτ (49 , , τ d m P ∆ h d mmf (50 , ,
28) ∆ h g τ h d e τ (54 , , h i M P h Cτ (54 , , τ ∆ h τ M c Lemma 5.13(55 , , τ gm ∆ h d d ( τ gm )(55 , , τ P d e P c d h (55 , , B τ h gn Cτ (56 , , h c e M h c Cτ (56 , , P h e M P h Cτ (56 , , gt τ (56 , , τ ∆ h d e P ∆ h c d d ( τ ∆ h d e )(57 , , τ e g c d e Cτ (57 , , h j M P h Cτ (57 , , τ P d m P ∆ h d mmf (57 , , Q τ gt τ ∆ h g (58 , , e g h gt Cτ (60 , , τ g M h c Cτ (61 , , D M h Cτ (62 , , τ ∆ h e g ∆ h c d Cτ (62 , ,
33) ∆ e ∆ h n Cτ (62 , , C ∆ h n Cτ (62 , , h · ∆ x + τ M e M P c Cτ (62 , , τ h · ∆ x h (62 , ,
32) ∆ h h (62 , , P i τ P h d tmf (63 , , h h ∆ h h +? τ M h e Cτ (63 , , τ P d e P c d h (64 , , τ P ∆ h d e P ∆ h c d d ( τ P ∆ h d e )(65 , ,
36) ∆ h m M P h c Cτ (65 , , τ P d m P ∆ h d mmf . TABLES 105 Table 6: Adams d differentials in the C -motivic Adams spectralsequence( s, f, w ) element d proof(67 , , τ Q + τ n h (67 , , C ′′ nm Cτ (67 , , X τ g (68 , , d h Q Cτ (68 , , τ h · ∆ g τ ∆ h e g Lemma 5.14(68 , , M h g τ (69 , , τ D ′ +? τ h G τ M h g Lemma 5.15(69 , , h B M h c d Cτ (69 , , τ ∆ h g τ e tmf (70 , , h h h p ′ Cτ (70 , , p τ h Q Cτ (70 , , τ M P e M P c d ( τ M P e )(70 , , m τ h e τ (71 , , τ P d e P c d h (72 , , τ P ∆ h d e P ∆ h c d d ( τ P ∆ h d e )(73 , , τ P d m P ∆ h d mmf (74 , , P h h τ h h Q Lemma 5.12(75 , , x , h x , Cτ (75 , , gB τ h g n h (75 , , τ g m ∆ h d e d ( τ g m )(76 , , h D d D Cτ (76 , ,
41) ∆ h h g h (76 , , g t τ (77 , , τ M h l ∆ h d Lemma 5.16(77 , , τ e g c e d ( τ e g )(77 , ,
41) ∆ h d τ d e tmf (78 , , h h h h h d Cτ (78 , , e g h g t Cτ (78 , , h x , τ e g Lemma 5.17(78 , , τ M P e M P c h (78 , , P i τ P h d tmf (79 , , x τ h m Lemma 5.18(79 , , τ P d e P c d h (79 , , P h Q ′ P h d Cτ (80 , ,
41) ∆ h h τ ∆ h e g +? τ ∆ h d e Lemma 5.20(80 , , τ d B ∆ h d e Lemma 5.21(80 , , τ P ∆ h d e P ∆ h c d d ( τ P ∆ h d e )(81 , , h h h , ,
42) ∆ p , , τ P d m P ∆ h d mmf (82 , , h Q h x , Cτ (82 , , P h h , , gC ∆ h gn Cτ (82 , , τ M e g M c d h
106 8. TABLES
Table 6: Adams d differentials in the C -motivic Adams spectralsequence( s, f, w ) element d proof(82 , , τ ∆ h e g ∆ h c d e d ( τ ∆ h e g )(83 , , τ h g + τ h e , ,
45) ∆ h e τ d e τ (84 , , f τ h h Q Cτ , h (84 , ,
45) ∆ c d τ ∆ h d e mmf (85 , , x , , , P ∆ h d τ d tmf (86 , , h c h (86 , , h h g ? τ h h Q (86 , , τ gG d (86 , ,
45) ∆ e ∆ h t Cτ (86 , , τ ∆ h e g ∆ h c d + τ e mmf (86 , , τ M P e M P c h (87 , , x , d (87 , , gC ′′ gnm Cτ (87 , , τ P d e P c d h (88 , ,
46) ∆ f h (88 , , τ M h d k P ∆ h d e Lemma 5.26(88 , ,
46) ∆ h d τ ∆ h d e +? P ∆ h d e mmf (88 , , τ P ∆ h d e P ∆ h c d d ( τ P ∆ h d e )(89 , , h h e M h g Cτ (89 , , h B g M h c e Lemma 5.27(89 , , τ ∆ h g τ e g τ (89 , , τ P d m P ∆ h d mmf (90 , , gm τ h e g τ (90 , ,
49) ∆ c e τ ∆ h d e mmf (92 , , x , d (92 , , mQ τ g n Cτ , τ (92 , , P ∆ c d τ P ∆ h d e mmf (93 , ,
48) ∆ h h τ h h d Lemma 5.29(93 , , x , ∆ h H Cτ (93 , , P h d , , τ M P h d j P ∆ h d Lemma 5.31(93 , , P ∆ h d τ P d tmf (94 , ,
49) ? x , ? h x , (94 , , τ h d e P h c d h (94 , ,
49) ∆ M h ? τ M d e (94 , , P h h e M ∆ h g Cτ (94 , , τ M d m M P ∆ h d d (94 , , τ ∆ h d e P ∆ h c d + τ d e mmf (94 , , τ M P e + M P c h + τ P ∆ h d (94 , , P i τ P h d tmf (95 , , g B τ h g n Cτ . TABLES 107 Table 6: Adams d differentials in the C -motivic Adams spectralsequence( s, f, w ) element d proof(95 , , M ∆ h g τ M h d e τ (95 , , τ g m ∆ h e + M h d e mmf (95 , ,
50) ∆ h c d τ d e m mmf (95 , , h · ∆ h i P h i Cτ (95 , , τ P d e P c d h Table 7: Adams d differentials in the C -motivic Adams spectralsequence( s, f, w ) element d proof(31 , , τ d e + h h P d Cτ (37 , , τ e g P d tmf (38 , , h h h x Cτ (39 , , τ P d e P d tmf (42 , , P h h d (47 , , τ P d e P d tmf (50 , , C P h h c Cτ (50 , , τ ∆ h g ij tmf (55 , , τ gm P ∆ c d + τ d j tmf (55 , , τ P d e P d tmf (56 , , τ ∆ h d e P ∆ h d tmf (57 , , τ e g d d (58 , , τ ∆ h d P ij tmf (62 , , τ h · ∆ x τ ∆ h d e Lemma 5.35(62 , ,
32) ∆ h , , τ ∆ h e g P ∆ h d tmf (63 , , C ′ M h d Cτ (63 , , τ X τ M h d Lemma 5.37(63 , , τ M h e M P h d (63 , , τ d m P ∆ c d + τ P d j tmf (63 , , h h P h i Cτ (64 , , τ ∆ h d g P ∆ h d tmf (66 , , τ P ∆ h d P ij tmf (68 , , h d X Lemma 5.38(69 , , τ h B M P h d d (70 , , h C ′′ h c Q Cτ (70 , , τ ∆ h g ∆ h d e τ (70 , , τ M P e M P d (71 , , τ P d m P ∆ c d + τ P d j tmf (71 , , τ P d e P d tmf (72 , , h G τ g n Lemma 5.40(72 , , τ P ∆ h d e P ∆ h d tmf
08 8. TABLES
Table 7: Adams d differentials in the C -motivic Adams spectralsequence( s, f, w ) element d proof(74 , , τ P ∆ h d P ij tmf (75 , , h d h x , Cτ (75 , ,
40) ∆ h h g τ M h d Lemma 5.41(75 , , τ g m ∆ c d + τ d l tmf (76 , , τ M d e M P d h (76 , ,
41) ∆ h h g τ ∆ h d e Lemma 5.42(77 , , τ e g d e τ (78 , , h h h h x , Cτ (79 , , τ P d m P ∆ c d + τ P d j tmf (79 , , τ P d e P d tmf (80 , , h e τ h x , Lemma 5.43(80 , , τ P ∆ h d e P ∆ h d tmf (81 , , τ gD h (81 , ,
42) ∆ h h τ d e l Lemma 5.44(82 , , τ M e g M P d h (82 , , τ ∆ h e g ∆ h d τ (82 , , τ P ∆ h d P ij tmf (83 , , gC ′ M h e g Cτ (83 , ,
45) ∆ j τ M h e g Lemma 5.45(86 , , h c τ h h h Q Lemma 5.48(86 , , h h i d , Cτ (86 , , τ M P e M P h (87 , , x , , , τ ∆ h H , ,
47) ∆ c g τ (87 , , τ P d m P ∆ c d + τ P d j tmf (87 , , τ P d e P d tmf (88 , ,
48) ∆ h g τ ∆ h d e mmf (88 , , τ P ∆ h d e P ∆ h d tmf (89 , , τ h B g M h d Lemma 5.51(90 , , h gC ′′ P h h c Cτ (90 , , τ gm ∆ h d e mmf (90 , , τ ∆ P h d P ij tmf (91 , ,
48) ∆ h A ′ , ,
50) ∆ h c e τ d e mmf (92 , , h gG τ g n τ (93 , , h h h g Lemma 5.53(95 , ,
50) ∆ M h ? M P ∆ h e (95 , , τ M ∆ h g M P ∆ h e d (95 , , τ g m ∆ c d e + τ d e m τ (95 , , τ P d m P ∆ c d + τ P d j tmf (95 , , P ∆ h i + τ P d e P d tmf . TABLES 109 Table 8: Adams d differentials in the C -motivic Adams spectralsequence( s, f, w ) element d proof(56 , , τ P h e τ ∆ h d e [ , Lemma 3.92](61 , , A ′ τ M h d [ , Theorem 12.1](63 , , τ h · ∆ x τ d e Lemma 5.55(63 , , h h P d Cτ (67 , , h Q + h D Cτ , τ (68 , , h d i τ ∆ h d Lemma 5.56(70 , , τ p + h h h τ h C ′ Lemma 5.57, [ ](72 , , h x , , , h D τ d g Lemma 5.59(81 , , gA ′ τ M h e τ (85 , , x , h (86 , , h h i ∆ h x Cτ (86 , , τ gG τ M ∆ h d Lemma 5.60(92 , , g h d Lemma 5.61(92 , ,
48) ∆ g ? τ ∆ h g (93 , , h · ∆ h h ∆ h g Cτ (93 , , e x , M ∆ h c d Lemma 5.62Table 9: Higher Adams differentials in the C -motivic Adams spec-tral sequence( s, f, w ) element r d r proof(67 , , τ Q + τ n , , h H M c d Lemma 5.66(68 , , τ h H M P d Lemma 5.66(77 , , m , , h x , , h h h , , τ h g + τ h e h n (85 , , x , M ∆ h d (86 , , h h g +? h f
10 ? M ∆ h d (87 , , x , τ M ∆ h e (87 , , gQ , , τ ∆ h H τ M ∆ h e (88 , , h gH M c e Lemma 5.74(88 , , τ h gH , ,
46) ∆ f τ M d (88 , ,
48) ∆ g g M d Lemma 5.75(92 , ,
51) ∆ h e , ,
51) ∆ h e , , τ e x , M P ∆ h d [ ]
10 8. TABLES
Table 10: Unknown Adams differentials in the C -motivic Adamsspectral sequence( s, f, w ) element r possible d r (63 , , h h h h +? τ M h e (69 , , D ′ h X +? τ h C ′′ (80 , ,
41) ∆ h h τ ∆ h e g +? τ ∆ h d e (83 , , τ h g + τ h e h n (85 , , x , M ∆ h d (86 , , h h g τ h h Q (86 , , h h g +? h f
10 ? M ∆ h d (87 , , x , τ M ∆ h e (87 , , τ ∆ h H τ M ∆ h e (88 , ,
46) ∆ f τ M d (88 , ,
46) ∆ h d τ ∆ h d e +? P ∆ h d e (92 , ,
48) ∆ g τ ∆ h g (94 , , x , τ h x , (94 , , x , h x , (94 , ,
49) ∆ M h τ M d e (95 , , x , x , +? τ d H (95 , ,
50) ∆ M h M P ∆ h e . T A B L E S Table 11: Some Toda brackets( s, w ) bracket contains indeterminacy proof used in(2 , h , η, i τ h h h , h , h i , h η, ν, η i h h h , h , h i , h ν, η, ν i h h h h , h , h i , h η, ν, ν i c h h h h , h , h h i , h η , ν, η, i c h h , h , h , h i , h , ǫ, i τ h c , h η, , σ i P h τ h h , τ h c h h , h , h h i , h , µ , i τ P h , h , η, τ ηǫ i P h P h h , τ P h h h , h , τ h c i , h , σ, σ i h h h h , h h d ( h ) = h h , h , κ, i τ h d h h , h h Corollary 6.2 7.37, 7.156 h h , h h (16 , h η, σ , i h h P c d ( h ) = h h , h η, , σ i h h P c d ( h ) = h h , h , η , i τ h h , h , η, ηκ i h d d ( e ) = h d , h ν, σ, σ i h h d ( h ) = h h , h κ, , η, ν i τ g τ h g Lemma 6.5 7.108(20 , h ν, η, ηκ i h g h g d ( e ) = h d , h ν, ν, κ i τ h g h h d ( f
0) = h e , h σ, , ρ i h i + τ P h d τ h c h h , h , h h i , h σ , , σ , i h d ( h ) = h h , h η, , θ i h h d ( h ) = h h , h η, σ , η, σ i d h h , h , h , h i , h ηθ , η, i p h h h , h , h i . T A B L E S Table 11: Some Toda brackets( s, w ) bracket detected by indeterminacy proof used in(36 , h τ, η κ , η i t P h Lemma 6.9 Table 5(36 , h ν, η, ηθ i t , ?] 7.22(39 , h η , ν, ν i h c h h h h h h , h , h h i = 7.147= h h h , h , h h i (39 , h ǫ, , θ i h c d ( h ) = h h , h , θ , κ i h d h h h , h h d d ( h ) = h h h h d (57 , h τ, τ ηκκ , η i h h h i P h c d ( τ P h e ) = τ ∆ h d e , h , θ , θ , i h h n d ( h ) = h h , h η, ηκ, τ θ . i ∆ e + C , h τ ηθ . , κ, , η i τ h H Remark 7.109 7.108(64 , h η, , θ i h h τ h h , τ h X d ( h ) = h h τ h Q , P c , h ν, η, τ κθ . i h A ′ d ( A ′ ) = τ M h d , h η , θ , η , θ i ∆ h Lemma 6.15 7.22(68 , h σ, κ, τ ηθ . i h A ′ d ( A ′ ) = τ M h d , h ν , , θ i h h d ( h ) = h h , h σ, , θ i h h h τ h p ′ d ( h ) = h h , h η, ν, τ θ . κ i τ h D ′ Lemma 6.17 7.26, 7.116(70 , h η, τ κ , τ κ i τ ∆ h g + d (∆ h ) = τ h d Table 5+ τ ∆ h g d ( τ ∆ h g ) = τ e (71 , h ǫ, , θ i h c d ( h ) = h h , h ν, ǫ, κθ . i h H τ M h g d ( h H ) = M c d , h µ , , θ i P h h d ( h ) = h h , h τ κθ . , ν, ν i h d D h h h d ( P ( A + A ′ )) = τ M h h g . T A B L E S Table 11: Some Toda brackets( s, w ) bracket detected by indeterminacy proof used in(72 , h σ , , { t } , τ κ i h Q + h D Lemma 6.19 7.32(75 , h τ ηκκ , η, η η i ∆ h d e h ∆ h d e , h , h h i , h σ , , θ i h h d ( h ) = h h d ( h ) = h h (77 , h κ, , θ i h d Lemma 6.21 7.37, Table 5(79 , h η , , θ i h h h d ( h ) = h h , h{ h x , } , , η i h h x , h h x , , h , h i = 7.39= x , h h , h , h i (80 , h µ , , θ i P h h d ( h ) = h h Table 5(80 , h , η, τ η { h x , }i τ h x Lemma 6.22 Table 5(82 , h σ, , θ i h c d ( h ) = h h , h{ ∆ e + C } , η , η i (∆ e + C ) g h x h ∆ e + C , h , h h i , h η κ, η, ν i h h g d ( h e ) = h h d Table 5, 7.91(84 , h νν , , θ i h h h d ( h ) = h h , h τ η κ, , θ i P h d d ( h ) = h h Table 5(87 , h θ , τ κ, { t }i τ gQ d ( Q ) = τ gt , h τ { h Q + h n } , ν , η i h c Lemma 6.26 7.50, Table 5(88 , h µ , , θ i P h h d ( h ) = h h Table 5(89 , h ν, η, { h gA ′ }i ∆ h g g τ h gC ′ Remark 7.139 7.138(91 , h{ h h g } , , θ i h h h g d ( h ) = h h Table 5(92 , h θ , θ , , θ i h g Lemma 5.76(93 , h θ , , θ i h h h Cτ Table 5(93 , h κ , κ, τ ηθ . i h x , d ( A ′ ) = τ M h d Table 5(94 , h{ n } , , θ i h n d ( h ) = h h Table 5(95 , h{ ∆ h h } , , θ i ∆ h h h d ( h ) = h h Table 5(95 , h κ , , θ i h d d ( h ) = h h Table 5
14 8. TABLES
Table 12: Hidden values of inclusion of the bottom cell into Cτ ( s, f, w ) source value proof(50 , , τ C P h h c (57 , , h h h i ∆ h h (63 , , τ h H h B (63 , , τ X + τ C ′ h d e (64 , , τ h X h h d e (66 , , τ h C ′ τ B (70 , , τ h h H h h B (70 , , τ h D ′ h X (70 , , h h (∆ e + C ) + τ h C ′′ h c Q Lemma 7.7(71 , , τ h p h h Q (74 , , h ( τ Q + τ n ) h p ′ (75 , , h h Q τ h g n (77 , , τ h h D x , (81 , , τ gD h x , (83 , , h c A ′ h gB (85 , , τ h f h c (86 , , τ h f τ h h Q (86 , , τ h h g +? τ h f ?∆ e + τ ∆ h e g (86 , , τ h x , ?∆ e + τ ∆ h e g (86 , , τ h gC ′ τ B g (87 , , τ ∆ h H ?∆ h B (88 , , τ h gH ∆ g g (90 , , τ h gC ′′ P h h c (90 , , τ gm ∆ c e Table 13: Hidden values of projection from Cτ to the top cell( s, f, w ) source value crossing source(30 , ,
16) ∆ h h d (34 , , h h h d (38 , , h y τ h e g (41 , , h c h h d (44 , ,
24) ∆ h d h d (50 , ,
28) ∆ h g h d e (55 , , B h gn (56 , ,
29) ∆ h h ∆ h d e (57 , , Q τ gt (58 , , h D ∆ h d (58 , , P h h e τ h e g (59 , , h D h d g (61 , , A ′ M h d h Q (62 , , P h c d τ ∆ h d e
0. TABLES 115
Table 13: Hidden values of projection from Cτ to the top cell( s, f, w ) source value crossing source(63 , , P h h c d τ d e (64 , , km h d e (65 , , h H d (65 , , h h D h (∆ e + C )(66 , , h h h h h (68 , , h d h · ∆ h (68 , , h H h X h h j (68 , , τ M c d τ ∆ h e g (68 , , h d i ∆ h d (69 , , h X τ M h g (69 , , P ( A + A ′ ) τ M h h g (69 , , τ ∆ h g τ e (70 , , h h h τ h C ′ (70 , , m h e (72 , , h G g n (72 , , d D τ M h g (73 , , h D τ d g (74 , , P h h h h Q (75 , , h d D M h d (75 , , gB h g n (76 , ,
41) ∆ h h g ∆ h d e (77 , , τ gQ τ g t (78 , , h x , h d g (78 , , h x , τ e g (80 , , h e h x , (80 , ,
41) ∆ h h τ ∆ h e g (81 , , gA ′ M h e P h h (81 , ,
42) ∆ h h τ d e l (82 , ,
44) ∆ e τ ∆ h e (83 , , τ h g ? τ (∆ e + C ) g P h h c (83 , ,
45) ∆ h e τ d e (84 , , f h h Q (84 , , d m h d e (85 , , h f h h Q (85 , , x , ?∆ h j (85 , , h gH d g (86 , , h c h h h Q (86 , , h f h h Q (86 , , h h g ? τ M h g (86 , , h x , ? τ M h g (86 , ,
45) ∆ e + τ ∆ h e g ? M ∆ h d P h h (86 , ,
44) ∆ h h τ ∆ h e g (87 , , x , ? M ∆ h e (87 , ,
45) ∆ h B ? M ∆ h e
016 8. TABLES
Table 13: Hidden values of projection from Cτ to the top cell( s, f, w ) source value crossing source(87 , ,
45) ∆ h d τ e m (88 , , h gH M h g (88 , , τ ∆ g τ ∆ h e g (88 , ,
48) ∆ h g ∆ h d e (89 , ,
46) ∆ h c ∆ h h d (89 , , τ ∆ h g τ e g (90 , , gm h e g Table 14: Hidden τ extensions in the C -motivic Adams spectralsequence( s, f, w ) from to proof(22 , , c d P d (23 , , h c d P h d (28 , , h h g d (29 , , h h g h d (40 , , τ h g ∆ h d (41 , , τ h g ∆ h e (42 , , c e d (43 , , h c e h d (46 , , h g ∆ h c (47 , ,
26) ∆ h c d P ∆ h d (48 , , h h g d e (49 , , h h g h d e (52 , ,
29) ∆ h h g τ e m (53 , , M c M P (53 , ,
30) ∆ h h g ∆ h d (54 , , h i M h c (54 , , M h c M P h (54 , , h h e τ e g (55 , , h h e τ h e g (55 , , τ h e g ∆ h d e (59 , , j M d (59 , , P h h e τ ∆ h d g (60 , , h D M h d (60 , , τ h g ∆ c d + τ d l (61 , , τ h g ∆ h d e (62 , , h h c e d e (63 , , h h g h d e (65 , , h X τ M g (66 , , h X τ M h g (66 , , P h h c e τ d e m (67 , , P h h c e ∆ h d
30. TABLES 117
Table 14: Hidden τ extensions in the C -motivic Adams spectralsequence( s, f, w ) from to proof(68 , , h h g e (69 , , h h g h e (70 , , d e h h (∆ e + C ) Lemma 7.7(70 , , τ h C ′′ + h h (∆ e + C ) ∆ h c (71 , , h H h Q (72 , , h x , h d D (72 , ,
41) ∆ h h g τ e gm (73 , , h h c h h D (73 , , τ h C ′′ ∆ h h c (73 , , M h h g M d (73 , ,
42) ∆ h h g ∆ h d e (74 , , M h h g M h d (75 , , τ h e g ∆ h d e (77 , ,
42) ∆ h g τ e g (78 , , h m M ∆ h h (79 , , gj M e (80 , , h gj M h e (80 , , τ h g ∆ h e g (80 , ,
46) ∆ h e g ∆ c d e + τ d e l (81 , , τ h g ∆ h e (82 , , τ (∆ e + C ) g ?∆ h n (82 , , c e g d e (83 , , h c e g h d e (84 , ,
46) ∆ h j ? M ∆ h d (85 , , h c d P h d (85 , , τ M h g ? M ∆ h d (86 , , h h c d P h h d (86 , , τ h gC ′ ∆ h d (86 , , τ M h g M ∆ h e (87 , ,
50) ∆ h c e g ∆ h d e (88 , , h h g e g + M h e g (89 , , τ h gC ′ ∆ h h d (89 , , h h g h e g + M h e g (90 , , τ h gC ′′ ∆ h c g
18 8. TABLES
Table 15: Hidden 2 extensions( s, f, w ) source target proof notes(23 , , τ h h g P h d τ (23 , , h h g h c d Cτ (40 , , τ g ∆ h d τ (43 , , τ h h g h d τ (43 , , h h g h c e Cτ (47 , , τ ∆ h e P ∆ h d τ (47 , ,
26) ∆ h e ∆ h c d Cτ (51 , , h h g τ gn [ ](54 , , h h i τ e g Lemma 7.19, [ ](60 , , τ g ∆ c d + τ d l τ (63 , , τ h H τ h (∆ e + C ) Lemma 7.20(63 , , τ h h g h d e τ (64 , , h h τ h h Lemma 7.21(65 , , h X M h g τ (67 , , τ ∆ h e g ∆ h d τ (70 , , τ h h H τ h h (∆ e + C ) Lemma 7.20(71 , , h c τ h p ′ Lemma 7.28(71 , , h H τ M h g Lemma 7.30(74 , , h ( τ Q + τ n ) τ x , Lemma 7.36 indet(74 , , h C ′′ M h d Cτ (74 , ,
40) ∆ h g τ e g mmf (77 , , τ h h D h x , Cτ (78 , , e A ′ M ∆ h h Lemma 7.38(80 , , τ g ∆ c d e + τ d e l τ (80 , , τ g ∆ h e g + M h d e τ (83 , , τ h h g h d e τ (83 , , h h g h c e g Cτ (85 , , τ h f τ h h Q Lemma 7.45(85 , , τ M g ? M ∆ h d τ (86 , , τ h h h g P h h d τ (86 , , h h h g h h c d Cτ (87 , , gQ B d Cτ (87 , ,
45) ? τ ∆ h H ? τ ∆ h d Lemma 7.51? τ ∆ c g (87 , , τ ∆ h e g ∆ h d e Cτ (87 , ,
50) ∆ h e g ∆ h c e g + M h c d e Cτ (90 , , h gQ τ d e g Cτ . TABLES 119 Table 16: Possible hidden 2 extensions( s, f, w ) source target(59 , , j ? τ c g (72 , , h x , ? τ d g (79 , , gj ? τ c g (82 , , h g ? τ (∆ e + C ) g (82 , , τ h x , ?∆ h n (82 , , h x , ? τ (∆ e + C ) g (85 , , x , ? τ P h x , (86 , ,
45) ? τ h x , ? P h h d ? τ ∆ h d ? τ M ∆ h e (87 , , x , ? τ gQ ? τ ∆ h d ? τ ∆ c g
20 8. TABLES
Table 17: Hidden η extensions( s, f, w ) source target proof notes(15 , , h h P c Cτ (21 , , τ h g P d τ (21 , , τ h g c d Cτ (23 , , h i P c Cτ (31 , , h h P c Cτ (38 , , h h h τ c g [ , Table 29](39 , , P h i P c Cτ (40 , , τ g ∆ h e τ (41 , , h f τ h c g [ , Table 29] crossing(41 , , τ h g d τ (41 , , τ h g c e Cτ (41 , ,
22) ∆ h e τ d τ (45 , , h h M h [ , Table 29] crossing(45 , , τ h g ∆ h c τ (45 , , τ ∆ h g τ d l + ∆ c d Cτ (46 , , τ d l P ∆ h d τ (47 , ,
26) ∆ h e τ d e mmf (47 , , h Q ′ P c Cτ (50 , , τ C τ gn [ ](52 , , τ e m ∆ h d τ (54 , , τ e g ∆ h d e τ (55 , , P h i P c Cτ (59 , , τ ∆ h d g ∆ c d + τ d l τ (60 , , τ g ∆ h d e τ (61 , , h j τ h c g Cτ crossing(61 , , τ h g d e τ (61 , ,
34) ∆ h d e τ d e τ (63 , , τ h H h Q Cτ indet(63 , , h h P c Cτ (64 , , τ h X c Q Cτ (64 , , τ h X τ M h g Lemma 7.66 indet(65 , , τ ∆ h g τ d e m τ (66 , , τ d e m ∆ h d τ (67 , ,
38) ∆ h e g τ e Cτ (68 , , h A ′ h (∆ e + C ) Lemma 7.70(69 , , h h τ h h Q Lemma 7.71 crossing(70 , , τ h h H h Q Lemma 7.72(70 , , τ h D ′ d Q Cτ (71 , , τ h p h Q Cτ (71 , ,
38) ∆ h g τ e gm mmf (71 , , P h i P c Cτ (72 , , τ h h c τ h Q Lemma 7.78 indet(72 , , h d D τ M d Lemma 7.80 indet(72 , , τ e gm ∆ h d e τ . TABLES 121 Table 17: Hidden η extensions( s, f, w ) source target proof notes(74 , , τ e g ∆ h d e τ (75 , , h h d τ d g Lemma 7.81(75 , , h x , h gB Cτ (75 , , h h Q τ g t Cτ (76 , , x , M ∆ h h Cτ (77 , , τ h h D x , Cτ crossing(78 , , h h h τ ∆ B Cτ , [ ](78 , , e A ′ τ M e [ ](79 , , τ ∆ h e g ∆ c d e + τ d e l τ (79 , , P h Q ′ P c Cτ (80 , , τ g ∆ h e τ (81 , , h gj τ h c g Cτ crossing(81 , , τ h g d e τ (81 , , τ h g c e g Cτ (81 , ,
46) ∆ h e τ d e τ (83 , , τ ∆ j + τ gC ′ ? M ∆ h d τ crossing(84 , , τ h h g P h d τ (84 , , τ h h g h c d Cτ (85 , , τ M g M ∆ h e τ (85 , , τ ∆ h g ∆ c e g + M h c d e Cτ + τ e m (86 , , h h i τ ∆ c g Lemma 7.95(86 , , P h h τ ∆ h e g Cτ (86 , , τ e m ∆ h d e τ (87 , , h x , x , Cτ (87 , ,
45) ? τ ∆ h H ∆ f +? τ ∆ g g Cτ (87 , ,
50) ∆ h e g τ e g Cτ (87 , , P h i P c Cτ (88 , , τ h gH ∆ h g g + τ h gC ′ Cτ (89 , ,
47) ∆ h f τ ∆ h c g Lemma 7.98 crossing
22 8. TABLES
Table 18: Possible hidden η extensions( s, f, w ) source target proof(64 , , h D ? τ k (66 , , τ ∆ h ? τ ∆ h e g (66 , ,
35) ∆ h h ? τ ∆ h e g (67 , , h Q + h D ? τ M h h g (81 , , h h h ? τ h g Lemma 7.86? τ e g ?∆ h n (81 , , h h h ? τ (∆ e + C ) g Lemma 7.87(81 , , τ gD ?∆ h n (81 , , τ gD ? τ (∆ e + C ) g (86 , , τ h f ? τ gQ ?∆ h d (86 , , τ h h g +? τ h f ? τ gQ Lemma 7.93?∆ h d (86 , , h h g +? h f ? h x , ? τ h gA ′ (86 , , τ h x , ? τ gQ ? Lemma 7.93?∆ h d ?(87 , , h c ? τ h g (87 , , τ h h h c ? τ h g (87 , , gQ ? τ M h h g (88 , , g ? τ h gC ′ ?∆ h g g (89 , , h h g ? τ h gQ
3. TABLES 123
Table 19: Hidden ν extensions( s, f, w ) source target proof notes(20 , , τ h g P h d τ (20 , , h g h c d Cτ (22 , , h c h h c Cτ (26 , , τ h g h d τ (30 , , h p Cτ (32 , ,
17) ∆ h h τ h e tmf (39 , ,
21) ∆ h d τ d tmf (40 , , τ h g h d τ (40 , , h g h c e Cτ (42 , , h c g h h c Cτ (45 , , h h M h Cτ crossing(45 , , h h h M h h Cτ crossing(45 , , τ ∆ h g τ d e mmf (46 , , τ h g h d e τ (48 , , h h d τ gn [ ] crossing(51 , , τ M h M P h τ (51 , , M h M h c [ , Table 31](52 , ,
29) ∆ h h g τ h e g τ (52 , , τ e m ∆ h d e mmf (53 , , i gt Cτ (54 , , h h e h e g τ (57 , , h h h i τ d l tmf (59 , , P h h e τ d e τ (59 , , τ ∆ h d g τ d e mmf (60 , , τ h g h d e τ (62 , ,
33) ∆ e + C τ M h g Lemma 7.106(62 , , h c g h D Cτ (63 , , τ h H τ M h g Lemma 7.108 crossing(65 , , h h τ h Q Cτ (65 , , h X M h g τ (65 , , τ ∆ h g + P h h c e τ e mmf (66 , ,
36) ∆ h h C ′ Cτ (66 , , τ h g h e τ (67 , , h A ′ h h (∆ e + C ) Lemma 7.112(68 , , P h h j ∆ h h g Cτ (69 , , h C ′ τ d g Lemma 7.115(70 , , τ h D ′ τ M d Lemma 7.116 indet(70 , ,
37) ∆ h c ∆ h h c τ (70 , , τ ∆ h g + τ m τ ∆ h d e mmf (71 , , h H h C ′′ Cτ (71 , , τ M h g M h d τ (71 , ,
38) ∆ h h g τ e g mmf (72 , ,
41) ∆ h h g τ h e g τ (72 , , τ e gm ∆ h d e mmf (73 , , h C ′′ τ g t Lemma 7.118
24 8. TABLES
Table 19: Hidden ν extensions( s, f, w ) source target proof notes(74 , , τ ∆ h g τ e g τ (77 , , h h τ h x Lemma 7.123 indet(77 , , h x , c A ′ Cτ (77 , ,
42) ∆ h g τ d e l τ (77 , , τ e g τ d e l mmf (78 , , h h h ∆ p Cτ (78 , , h h h τ ∆ h d Lemma 7.124(79 , ,
45) ∆ h e g + M h d e τ d e mmf (80 , , τ h g h d e τ (80 , , h g h c e g Cτ (82 , , h g h h h Q Cτ (82 , , h x , P h x , Cτ (82 , , P h h ∆ h x Cτ (82 , , τ (∆ e + C ) g ? M ∆ h d τ (82 , ,
45) (∆ e + C ) g τ M h g Lemma 7.130(82 , , h c g h h c Cτ (83 , , τ h h g P h h d τ (83 , , h h g h h c d Cτ (83 , , τ gC ′ M ∆ h e Lemma 7.132 crossing(83 , ,
45) ∆ j + τ gC ′ τ M h g Lemma 7.132(84 , , h gD B d Cτ (85 , , τ h f h x , + τ g Lemma 7.135(85 , , h h c h h h c Cτ (85 , , h h Q h g Cτ (85 , , τ ∆ h g τ e g mmf (86 , , τ h g h e g + M h e g τ (87 , , h gA ′ ∆ h g g Lemma 7.138Table 20: Possible hidden ν extensions( s, f, w ) source target(70 , , h h h ? h h D (75 , , h h d ? M ∆ h h (81 , , h h h h ? M ∆ h d (86 , , h h i ? τ ∆ h f (87 , , h c ? τ M ∆ h g (87 , , x , ? τ M ∆ h g (87 , , τ gQ ? τ M ∆ h g (87 , , τ ∆ h H ? τ M ∆ h g . TABLES 125 Table 21: Miscellaneous hidden extensions( s, f, w ) type source target proof(16 , , σ h h h c Cτ (20 , , ǫ τ g d [ , Table 33](30 , , σ h x Cτ (30 , , η h h h d [ , Table 33](32 , , ǫ ∆ h h ∆ h d tmf (32 , , κ ∆ h h τ d l + ∆ c d tmf (44 , , θ g x , Cτ (45 , , ǫ τ h h M P
Lemma 7.140(45 , , ǫ h h M c Lemma 7.140(45 , , κ h h M d Lemma 7.142(45 , , κ h h τ M g Lemma 7.144(45 , , { ∆ h h } h h M ∆ h h Lemma 7.145(45 , , θ . h h M Lemma 7.146(62 , , σ h p ′ Cτ (62 , , κ h h h A Lemma 7.149(62 , , ρ h ? h x , Lemma 7.150? τ m (62 , , θ h h g Cτ (63 , , ǫ τ X + τ C ′ d Q Cτ (63 , , κ τ X + τ C ′ M ∆ h h Cτ (63 , , η τ X + τ C ′ h x , Cτ (64 , , ρ h h P h c Lemma 7.151(64 , , ρ h h P h c Lemma 7.151(65 , , ǫ τ M g M d Lemma 7.152(69 , , σ p ′ h h A Cτ (77 , , ǫ M ∆ h h ? M ∆ h d Lemma 7.153(79 , , σ h h h h h c Cτ ibliography [1] J. F. Adams, On the non-existence of elements of Hopf invariant one , Ann. of Math. (2) (1960), 20–104. MR0141119 (25 Relations amongst Toda brackets andthe Kervaire invariant in dimension
62, J. London Math. Soc. (2) (1984), no. 3, 533–550,DOI 10.1112/jlms/s2-30.3.533. MR810962 (87g:55025)[3] M. G. Barratt, M. E. Mahowald, and M. C. Tangora, Some differentials in the Adams spectralsequence. II , Topology (1970), 309–316. MR0266215 (42 Detecting exoticspheres in low dimensions using coker J , J. London Math. Soc., to appear.[5] Robert Burklund, Daniel C. Isaksen, and Zhouli Xu, Synthetic stable stems , in preparation.[6] Robert Bruner,
A new differential in the Adams spectral sequence , Topology (1984), no. 3,271–276, DOI 10.1016/0040-9383(84)90010-7. MR770563 (86c:55016)[7] Robert R. Bruner, The cohomology of the mod 2 Steenrod algebra: A computer calculation ,Wayne State University Research Report (1997).[8] , Squaring operations in
Ext A ( F , F ) (2004), preprint.[9] Bogdan Gheorghe, The motivic cofiber of τ , Doc. Math. (2018), 1077–1127. MR3874951[10] Bogdan Gheorghe, Daniel C. Isaksen, Achim Krause, and Nicolas Ricka, C -motivic modularforms (2018), submitted, available at arXiv:1810.11050 .[11] Bogdan Gheorghe, Guozhen Wang, and Zhouli Xu, The special fiber of the motivic de-formation of the stable homotopy category is algebraic (2018), submitted, available at arXiv:1809.09290 .[12] Bertrand J. Guillou and Daniel C. Isaksen,
The η -local motivic sphere , J. Pure Appl. Algebra (2015), no. 10, 4728–4756, DOI 10.1016/j.jpaa.2015.03.004. MR3346515[13] M. A. Hill, M. J. Hopkins, and D. C. Ravenel, On the nonexistence of elements of Kervaireinvariant one , Ann. of Math. (2) (2016), no. 1, 1–262, DOI 10.4007/annals.2016.184.1.1.MR3505179[14] Daniel C. Isaksen,
The cohomology of motivic A (2), Homology Homotopy Appl. (2009),no. 2, 251–274. MR2591921 (2011c:55034)[15] , When is a fourfold Massey product defined? , Proc. Amer. Math. Soc. (2015),no. 5, 2235–2239, DOI 10.1090/S0002-9939-2014-12387-2. MR3314129[16] ,
Stable stems , Mem. Amer. Math. Soc. (2019), no. 1269, viii+159, DOI10.1090/memo/1269. MR4046815[17] ,
The homotopy of C -motivic modular forms , preprint, available at arXiv:1811.07937 .[18] , The Mahowald operator in the cohomology of the Steenrod algebra (2020), preprint.[19] Daniel C. Isaksen, Guozhen Wang, and Zhouli Xu,
Classical and motivic Adams charts (2020), preprint, available at s.wayne.edu/isaksen/adams-charts .[20] ,
Classical and motivic Adams-Novikov charts (2020), preprint, available at s.wayne.edu/isaksen/adams-charts .[21] ,
Classical algebraic Novikov charts and C -motivic Adams charts for the cofiber of τ (2020), preprint, available at s.wayne.edu/isaksen/adams-charts .[22] Daniel C. Isaksen and Zhouli Xu, Motivic stable homotopy and the stable 51 and 52 stems ,Topology Appl. (2015), 31–34, DOI 10.1016/j.topol.2015.04.008. MR3349503[23] Michel A. Kervaire and John W. Milnor,
Groups of homotopy spheres. I , Ann. of Math. (2) (1963), 504–537, DOI 10.2307/1970128. MR148075[24] Stanley O. Kochman, A chain functor for bordism , Trans. Amer. Math. Soc. (1978),167–196, DOI 10.2307/1997852. MR488031 [25] ,
Stable homotopy groups of spheres , Lecture Notes in Mathematics, vol. 1423,Springer-Verlag, Berlin, 1990. A computer-assisted approach. MR1052407 (91j:55016)[26] Stanley O. Kochman and Mark E. Mahowald,
On the computation of stable stems , The ˇCechcentennial (Boston, MA, 1993), Contemp. Math., vol. 181, Amer. Math. Soc., Providence,RI, 1995, pp. 299–316, DOI 10.1090/conm/181/02039. MR1320997 (96j:55018)[27] Achim Krause,
Periodicity in motivic homotopy theory and over BP ∗ BP , Ph.D. thesis, Uni-versit¨at Bonn, 2018.[28] Wen-Hsiung Lin, A proof of the strong Kervaire invariant in dimension 62 , First InternationalCongress of Chinese Mathematicians (Beijing, 1998), AMS/IP Stud. Adv. Math., vol. 20,Amer. Math. Soc., Providence, RI, 2001, pp. 351–358. MR1830191[29] Mark Mahowald and Martin Tangora,
Some differentials in the Adams spectral sequence ,Topology (1967), 349–369. MR0214072 (35 Matric Massey products , J. Algebra (1969), 533–568. MR0238929 (39 SOME ALGEBRAIC ASPECTS OF THE ADAMS-NOVIKOVSPECTRAL SEQUENCE , ProQuest LLC, Ann Arbor, MI, 1975. Thesis (Ph.D.)–PrincetonUniversity. MR2625232[32] John Milnor,
Differential topology forty-six years later , Notices Amer. Math. Soc. (2011),no. 6, 804–809. MR2839925[33] R. Michael F. Moss, Secondary compositions and the Adams spectral sequence , Math. Z. (1970), 283–310. MR0266216 (42
Methods of algebraic topology from the point of view of cobordism theory , Izv.Akad. Nauk SSSR Ser. Mat. (1967), 855–951 (Russian). MR0221509[35] Piotr Pstragowski, Synthetic spectra and the cellular motivic category (2018), preprint, avail-able at arXiv:1803.01804 .[36] Douglas C. Ravenel,
Complex cobordism and stable homotopy groups of spheres , Pure and Ap-plied Mathematics, vol. 121, Academic Press, Inc., Orlando, FL, 1986. MR860042 (87j:55003)[37] Martin C. Tangora,
On the cohomology of the Steenrod algebra , Math. Z. (1970), 18–64.MR0266205 (42
Composition methods in homotopy groups of spheres , Annals of MathematicsStudies, No. 49, Princeton University Press, Princeton, N.J., 1962. MR0143217 (26
Topological modular forms , Mathematical Surveys and Monographs, vol. 201, AmericanMathematical Society, Providence, RI, 2014. Edited by Christopher L. Douglas, John Francis,Andr´e G. Henriques and Michael A. Hill. MR3223024[40] Vladimir Voevodsky,
Motivic cohomology with Z / -coefficients , Publ. Math. Inst. Hautes´Etudes Sci. (2003), 59–104, DOI 10.1007/s10240-003-0010-6. MR2031199 (2005b:14038b)[41] , Motivic Eilenberg-Maclane spaces , Publ. Math. Inst. Hautes ´Etudes Sci. (2010),1–99, DOI 10.1007/s10240-010-0024-9. MR2737977 (2012f:14041)[42] Guozhen Wang, github.com/pouiyter/morestablestems .[43] ,
Computations of the Adams-Novikov E -term (2020), preprint, available at github.com/pouiyter/MinimalResolution .[44] Guozhen Wang and Zhouli Xu, The triviality of the 61-stem in the stable homotopy groupsof spheres , Ann. of Math. (2) (2017), no. 2, 501–580, DOI 10.4007/annals.2017.186.2.3.MR3702672[45] ,
Some extensions in the Adams spectral sequence and the 51–stem , Algebr. Geom.Topol. (2018), no. 7, 3887–3906, DOI 10.2140/agt.2018.18.3887. MR3892234[46] Zhouli Xu, The strong Kervaire invariant problem in dimension 62 , Geom. Topol.20