Some extensions in the Adams spectral sequence and the 51-stem
aa r X i v : . [ m a t h . A T ] D ec SOME EXTENSIONS IN THE ADAMS SPECTRAL SEQUENCE AND THE51-STEM
GUOZHEN WANG AND ZHOULI XU
Abstract.
We show a few nontrivial extensions in the classical Adams spectral sequence. Inparticular, we compute that the 2-primary part of π is Z / ⊕ Z / ⊕ Z /
2. This was the lastunsolved 2-extension problem left by the recent works of Isaksen and the authors ([5], [7], [20])through the 61-stem.The proof of this result uses the RP ∞ technique, which was introduced by the authors in[20] to prove π = 0. This paper advertises this technique through examples that have simplerproofs than in [20]. Contents
1. Introduction 1Acknowledgement 22. the 51-stem and some extensions 23. The method and notations 54. the σ -extension on h d
65. A lemma for extensions in the Atiyah-Hirzebruch spectral sequence 126. Appendix 13References 151.
Introduction
The computation of the stable homotopy groups of spheres is both a fundamental and a difficultproblem in homotopy theory. Recently, using Massey products and Toda brackets, Isaksen [5]extended the 2-primary Adams spectral sequence computations to the 59-stem, with a few 2 , η, ν -extensions unsettled.Based on the algebraic Kahn-Priddy theorem [10, 11], the authors [20] compute a few differentialsin the Adams spectral sequence, and proved that π = 0. The 61-stem result has the geometricconsequence that the 61-sphere has a unique smooth structure, and it is the last odd dimensionalcase. In the article [20], it took the authors more than 40 pages to introduce the method and proveone Adams differential d ( D ) = B . Here B and D are certain elements in the 60 and 61-stem.Our notation will be consistent with [5] and [20].In this paper, we show that our technique can also be used to solve extension problems in theAdams spectral sequence. We establish a nontrivial 2-extension in the 51-stem, together with a fewother extensions left unsolved by Isaksen [5]. As a result, we have the following proposition. Proposition 1.1.
There is a nontrivial 2-extension from h h g to gn in the 51-stem. We’d like to point out that this is also a nontrivial 2-extension in the Adams-Novikov spectralsequence.Combining with Theorem 1.1 of [7], which describes the group structure of π up to this 2-extension, we have the following corollary. Corollary 1.2.
The 2-primary π is Z / ⊕ Z / ⊕ Z / , generated by elements that are detected by h g , P h and h B . Using a Toda bracket argument, Proposition 1.1 is deduced from the following σ -extension inthe 46-stem. Proposition 1.3. (1) There is a nontrivial σ -extension from h d to N in the 46-stem.(2) There is a nontrivial η -extension from h g to N in the 46-stem. As a corollary, we prove a few more extensions.
Corollary 1.4. (1) There is a nontrivial η -extension from C to gn in the 51-stem.(2) There is a nontrivial ν -extension from h h d to gn in the 51-stem.(3) There is a nontrivial σ -extension from h g to gn in the 51-stem. In particular, the element gn detects σ θ . Remark 1.5.
In [5], Isaksen had an argument that implies the nonexistence of the two η -extensionson h g and C , which is contrary to our results in Proposition 1.3 and Corollary 1.4. Isaksen’sargument fails because of neglect of the indeterminacy of a certain Massey product in a subtle way.For more details, see Remark 2.3.The proof of the σ -extension in Proposition 1.3 is the major part of this article: we prove it bythe RP ∞ technique as a demonstration of the effectiveness of our method.The rest of this paper is organized as the following.In Section 2, we deduce Proposition 1.1 and Corollary 1.4 from Proposition 1.3. We also showthe two statements in Proposition 1.3 are equivalent. In Section 3, we recall a few notations from[20]. We also give a brief review of how to use the RP ∞ technique to prove differentials and tosolve extension problems. In Section 4, we present the proof of Proposition 1.3. In Section 5, weprove a lemma which is used in Section 4. The lemma gives a general connection that relates Todabrackets and extension problems in 2 cell complexes. In the Appendix, we use cell diagrams asintuition for statements of the lemmas in Section 5. Acknowledgement.
We thank the anonymous referee for various helpful comments. This materialis based upon work supported by the National Science Foundation under Grant No. DMS-1810638.2. the 51-stem and some extensions
We first establish the following lemma.
Lemma 2.1.
In the Adams E -page, we have the following Massey products in the 46-stem: gn = h N, h , h i = h N, h , h i Proof.
By Bruner’s computation [4], there is a relation in bidegree ( t − s, s ) = (81 , gnr = mN. OME EXTENSIONS IN THE ADAMS SPECTRAL SEQUENCE AND THE 51-STEM 3
We have
Ext , = Z / ⊕ Z /
2, generated by gnr and h x , . Moreover, the element gnr isnot divisible by h , and neither of the generators is divisible by h .By Tangora’s computation [16], we have a Massey product in the Adams E -page, m = h r, h , h i . Therefore, gn · r = m · N = N · h r, h , h i = h N · r, h , h i = r · h N, h , h i with zero indeterminacy. This implies gn = h N, h , h i . Because of the relation h · N = 0 in Ext , = 0, we also have gn · r = m · N = h r, h , h i · N = r · h h , h , N i . This implies gn = h N, h , h i . (cid:3) Based on Proposition 1.3, we prove part (1) of Corollary 1.4.
Proof.
By Proposition 1.3, N detects the homotopy class σ { d } . Then the Massey product gn = h N, h , h i and Moss’s theorem [13] imply that gn detects a homotopy class that is contained in the Todabracket h σ { d } , ν, η i . The indeterminacy of this Toda bracket is η · π + σ { d } · π = η · π , since π = 0. Shuffling this bracket, we have h σ { d } , ν, η i ⊇ σ { d } · h σ, ν, η i = 0 , since h σ, ν, η i ⊆ π = 0.Therefore, gn detects a homotopy class that lies in the indeterminacy, and hence is divisible by η . For filtration reasons, the only possibilities are C and h c . However, Lemma 4.2.51 of [5] statesthat there is no η -extension from h c to gn . Therefore, we must have an η -extension from C to gn . (cid:3) Based on Proposition 1.3, we prove part (2) of Corollary 1.4.
Proof.
By Proposition 1.3, N detects the homotopy class σ { d } . Then the Massey product gn = h N, h , h i and Moss’s theorem [13] imply that gn detects a homotopy class that is contained in the Todabracket h σ { d } , η, ν i . The indeterminacy of this Toda bracket is ν · π + σ { d } · π = ν · π , GUOZHEN WANG AND ZHOULI XU since π = 0. Shuffling this bracket, we have h σ { d } , η, ν i ⊇ σ · h σ { d } , η, ν i = σ { d } · h η, ν, σ i = 0 , since h η, ν, σ i ⊆ π = 0.Therefore, gn detects a homotopy class that lies in the indeterminacy, and hence is divisible by ν . For filtration reasons, the only possibility is h h d , which completes the proof. (cid:3) Now we prove part (3) of Corollary 1.4, and Proposition 1.1.
Proof.
Lemma 4.2.31 from Isaksen’s computation [5] states that the 2-extension from h h g to gn is equivalent to the ν -extension from h h d to gn . This proves Proposition 1.1.It is clear that Proposition 1.1 is equivalent to part (3) of Corollary 1.4, since σ is detected by h , and σ θ is detected by h g . (See [2, 5] for the second fact.) (cid:3) In the following Lemma 2.2, we show that the two statements in Proposition 1.3 are equivalent.
Lemma 2.2.
There is a σ -extension from h d to N if and only if there is an η -extension from h g to N .Proof. First note that there are relations in
Ext : h d = h e , h e = h g . By Bruner’s differential [3, Theorem 4.1] d ( e ) = h t = h n, we have Massey products in the Adams E -page h d = h e = h h n, h , h i , h g = h e = h h , h n, h i . Then Moss’s theorem implies that they converge to Toda brackets h ν { n } , ν, η i , h σ, ν { n } , ν i . Therefore, the lemma follows from the shuffling σ · h ν { n } , ν, η i = h σ, ν { n } , ν i · η. (cid:3) We give a remark on the two η -extensions we proved. Remark 2.3.
In Lemma 4.2.47 and Lemma 4.2.52 of [5], Isaksen showed that there are no η -extensions from h g to N or from C to gn . Both arguments are based the statement of Lemma3.3.45 of [5], whose proof implicitly studied the following motivic Massey product h h , P h h c , c i ∋ P h h e in the 59-stem of the motivic Adams E -page, which therefore converges to a motivic Toda bracket.However, in the motivic Adams E -page, the element P h h e is in the indeterminacy of this motivicMassey product, since P h h e is present in the motivic E -page (it supports a d differential).Therefore, we have h h , P h h c , c i = { P h h e , } instead. The statement of Moss’s theorem gives us the convergence of only one permanent cycle inthe motivic Massey product, therefore, in this case, it is inconclusive. OME EXTENSIONS IN THE ADAMS SPECTRAL SEQUENCE AND THE 51-STEM 5 The method and notations
In this section, we recall a few notations from [20] and set up terminology that will be used inSection 4.
Notation 3.1.
All spectra are localized at the prime 2. Suppose Z is a spectrum. Let Ext ( Z )denote the Adams E -page Ext A ( F , H ∗ ( Z ; F )) that converges to the 2-primary homotopy groupsof Z . Here A is the mod 2 dual Steenrod algebra.We now introduce some spectral sequence terminologies. A permanent cycle is a class that doesnot support any nontrivial differential. A surviving cycle is a permanent cycle that is also not thetarget of any differential.For spectra, let S be the sphere spectrum, and P ∞ be the suspension spectrum of RP ∞ . Ingeneral, we use P n + kn to denote the suspension spectrum of RP n + k /RP n − .Let α be a class in the stable homotopy groups of spheres. We use Cα to denote the cofiber of α . Definition 3.2.
Let A , B , C and D be CW spectra, i and q be maps A (cid:31) (cid:127) i / / B, B q / / / / C We say that (
A, i ) is an H F -subcomplex of B , if the map i induces an injection on mod 2 homology.We denote an H F -subcomplex by a hooked arrow as above.We say that ( C, q ) is an H F -quotient complex of B , if the map q induces a surjection on mod2 homology. We denote an H F -quotient complex by a double headed arrow as above.When the maps involved are clear in the context, we also say A is an H F -subcomplex of B , and C is an H F -quotient complex of B .Furthermore, we say D is an H F -subquotient of B , if D is an H F -subcomplex of an H F -quotient complex of B , or an H F -quotient complex of an H F -subcomplex of B .One example is that S , S , S are H F -subcomplexes of P ∞ , due to the solution of the Hopfinvariant one problem. Another example is that Σ Cη is an H F -subcomplex of P .We use the following way to denote the elements in the Adams E -page of P ∞ and its H F -subquotients. One way to compute Ext ( P ∞ ) is to use the algebraic Atiyah-Hirzebruch spectralsequence. E = L ∞ n =1 Ext ( S n ) + Ext ( P ∞ ) Notation 3.3.
We denote any element in
Ext ( S n ) by a [ n ], where a ∈ Ext ( S ), and n indicatesthat it comes from Ext ( S n ). We will abuse notation and write the same symbol a [ n ] for anelement of Ext ( P ∞ ) detected by the element a [ n ] of the Atiyah-Hirzebruch E ∞ -page. Thus, thereis indeterminacy in the notation a [ n ] that is detected by Atiyah-Hirzebruch E ∞ elements in lowerfiltration. When a [ n ] is the element of lowest Atiyah-Hirzebruch filtration in the Atiyah-Hirzebruch E ∞ -page in a given bidegree ( s, t ), then a [ n ] also is a well-defined element of Ext ( P ∞ ).We use similar notations for homotopy classes. Remark 3.4.
In [21], we computed differentials in the algebraic Atiyah-Hirzebruch spectral se-quence that converges to the Adams E -page of P ∞ in the range of t < P ∞ , one can also read offinformation about Ext ( P n + kn ). For details, see [21]. GUOZHEN WANG AND ZHOULI XU
Remark 3.5.
Despite the indeterminacy in Notation 3.3, there is a huge advantage of it. Suppose f : Q → Q ′ is a map between two H F -subquotients of P ∞ , and there exists an element a [ n ] whichis a generater of both Ext s,t ( Q ) and Ext s,t ( Q ′ ) for some bidegree ( s, t ) (this implies both Q and Q ′ have a cell in dimension n ). We must have that, with the right choices, a [ n ] in Ext s,t ( Q ) mapsto a [ n ] in Ext s,t ( Q ′ ). This property follows from the naturality of the algebraic Atiyah-Hirzebruchspectral sequence. L i ∈ I Ext ( S i ) (cid:11) (cid:19) / / L i ∈ I ′ Ext ( S i ) (cid:11) (cid:19) Ext ( Q ) / / Ext ( Q ′ ) a [ n ] ✤ / / a [ n ]4. the σ -extension on h d In this section, we prove part (1) of Proposition 1.3. The proof can be summarized in thefollowing “road map” with 4 main steps: S S (cid:127) _ (cid:15) (cid:15) σ o o ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ (cid:2) (cid:2) ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ ✝ N h d [7] P ∞ O O Σ Cη (cid:127) _ (cid:15) (cid:15) h t [9] ❴ O O h · h n [9] = h t [9] ❴ (cid:15) (cid:15) P ?(cid:31) O O / / / / P h t [9] ❴ O O ✤ / / h t [9]Here the elements in the right side of the “road map” are elements in the 46-stem of the E ∞ -pageof the Adams spectral sequences of the spectra in the corresponding positions.(1) Step 1 : We show that the element h t [9] is a permanent cycle in the Adams spectralsequence of Σ Cη , and hence a permanent cycle in the Adams spectral sequence of P .This is stated as Proposition 4.3.(2) Step 2 : Under the inclusion map S ֒ → Σ Cη , we show that the element h t [9] detectsthe image of σ { d } [7] in π (Σ Cη ). By naturality, the same statement is true, after wefurther map it to π ( P ). This is stated as Proposition 4.6. OME EXTENSIONS IN THE ADAMS SPECTRAL SEQUENCE AND THE 51-STEM 7 (3)
Step 3 : Under the any map S ֒ → P lifting the inclusion S ֒ → P , we show that theelement h t [9] in Ext ( P ) detects the image of σ { d } [7] in π ( P ). This is stated asProposition 4.7.(4) Step 4 : Using the inclusion map P → P ∞ and the transfer map P ∞ → S , we pushforward the element h t [9] in the E ∞ -page of P to the element N in the E ∞ -page of S .Since the composition S (cid:31) (cid:127) / / P / / P ∞ / / S is just σ , we have the desired σ -extension from h d to N in the Adams spectral sequencefor S . Remark 4.1.
Step 2 is the essential step. Intuitively, it comes from the zigzag of the following twodifferentials: d ( e ) = h t in the Adams spectral sequence of S , and d ( e [9]) = h e [7] = h d [7]in the algebraic Atiyah-Hirzebruch spectral sequence of Σ Cη that comes from the η -attaching map.Here h e = h d is a relation in Ext . This zigzag suggested that we consider the possibility that h t [9] detects σ { d } [7] in π (Σ Cη ).We start with Step 1. Proposition 4.3 is a consequence of the following lemma. Lemma 4.2.
The element h n [9] is a permanent cycle in Σ Cη , which detects a homotopy classthat maps to ν { n } [9] under the quotient map Σ Cη / / / / S . Proof.
The cofiber sequence S (cid:31) (cid:127) i / / Σ Cη p / / / / S η / / S gives us a long exact sequence of homotopy groups π ( S ) i ∗ / / π (Σ Cη ) p ∗ / / π ( S ) η / / π ( S ) . Since h n detects ν { n } , and η · ν { n } = 0 , there is an element α in π (Σ Cη ) such that p ∗ α = ν { n } [9].The element h n [9] in Ext ( S ) has Adams filtration 6, therefore by naturality, if it were notdetected by h n [9] in Ext (Σ Cη ), it would be detected by an element with Adams filtration atmost 5.From the same cofiber sequence, we have a short exact sequence on cohomology0 / / H ∗ ( S ) p ∗ / / H ∗ (Σ Cη ) i ∗ / / H ∗ ( S ) / / Ext groups
Ext s − ,t − ( S ) h / / Ext s,t ( S ) i ♯ / / Ext s,t (Σ Cη ) p ♯ / / Ext s,t ( S ) . GUOZHEN WANG AND ZHOULI XU
This gives us the Adams E -page of Σ Cη in the 42 and 43 stems for s ≤ Table 1.
The Adams E -page of Σ Cη in the 42 and 43 stems for s ≤ s \ t − s
42 436 h n [9] t [7]5 h p [9] h d [7]4 p [9] h h h [9]3 h h h [9]2 h h [9]The element h h [9] must support a nontrivial differential, since its image p ♯ ( h h [9]) supportsa d differential that kills h p [9] in the Adams spectral sequence of S .The elements h h h [9] and h h h [9] survive and detect homotopy classes that map to { h h h } [9]and { h h h } [9] in π ( S ). In fact, since there is no η -extension on h h h and h h h , we canchoose homotopy classes in π ( S ), which are detected by h h h [9] and h h h [9] and are zeroafter multiplying by η . Therefore, they have nontrivial pre-images under the map p ∗ in the longexact sequence of homotopy groups. For filtration reasons, their pre-images must be detected by h h h [9] and h h h [9] in the Adams spectral sequence of Σ Cη .Therefore, the only possibility left is h n [9], which completes the proof. (cid:3) We prove Proposition 4.3 in Step 1.
Proposition 4.3.
The elements h n [9] and h t [9] are permanent cycles in the Adams spectralsequence of Σ Cη , and hence also in that of P .Proof. We have a relation in
Ext : h · h n = h t. Therefore, h t [9] is product of permanent cycles. The second claim follows from the naturality ofthe Adams spectral sequences. (cid:3) For Step 2, we first show the following lemma.
Lemma 4.4.
The element h t [9] is not a boundary in the Adams spectral sequences of Σ Cη and P .Proof. The element h t [9] is hit by a d differential on e [9]In the Adams spectral sequence of S , we have the Bruner differential d ( e [9]) = h t [9] . However, the element e [9] is not present in either Ext (Σ Cη ) or Ext ( P ).Therefore, by naturality, the element h t [9] cannot be hit by any d r differential for r ≤ Cη and P .We have the Adams E -page of Σ Cη and P in the 46 and 47 stems for s ≤ h h h [9] and h h h [8].In the Adams spectral sequence of S , we have a d differential: d ( h h h [9]) = h x [9] . OME EXTENSIONS IN THE ADAMS SPECTRAL SEQUENCE AND THE 51-STEM 9
Table 2.
The Adams E -page of Σ Cη and P in the 46 and 47 stems for s ≤ Ext (Σ Cη ) Ext ( P ) s \ t − s
46 47 46 477 h t [9] • h t [9] • h x [9] h x [9] • h x [9] • h x [9] •• h x [8] • • • •• • • • • •• • •• • h h h [9] • h h h [9] h h h [8]By naturality of the quotient map to S , the element h h h [9] cannot support a d differentialthat kills h t [9].For the element h h h [8], it is straightforward to check it is a permanent cycle in the Adamsspectral sequence of P , and hence a permanent cycle in that of P . This rules out the candidate h h h [8] and completes the proof. (cid:3) Remark 4.5. In Ext , ( P ), the element h x [8] is clearly a surviving cycle. There are twopossibilities for the other element h x [9]: it is either killed by a d differential from h h h [9], or itsurvives and detects { h h h } [7]. We will leave the reader to figure out which way it goes.We prove Proposition 4.6 in Step 2. Proposition 4.6.
Under the inclusion map S ֒ → Σ Cη , the element h t [9] detects the imageof σ { d } [7] in π (Σ Cη ) . By naturality, the same statement is true after we further map it to π ( P ) .Proof. By Lemma 4.2 and Proposition 4.3, the element h n [9] survives in the Adams spectralsequence of Σ Cη , and detects a homotopy class that maps to ν { n } [9] under the quotient mapΣ Cη / / / / S . By Lemma 4.4, the element h t [9] = h · h n [9] survives and detects the homotopy class ν { n } [9] · ν .As showed in the proof of Lemma 2.2, the element h d = h e detects an element in the Todabracket h η, ν { n } , ν i . Therefore, by Lemma 5.3, we have ν { n } [9] · ν = h η, ν { n } , ν i [7] = σ { d } [7]in π (Σ Cη ). (cid:3) Now we prove Step 3.
Proposition 4.7.
Under the inclusion map S ֒ → P , the element h t [9] in Ext ( P ) detects theimage of σ { d } [7] in π ( P ) . The idea of the proof of Proposition 4.7 is to make use of naturality of the Adams filtrations. S (cid:31) (cid:127) / / P / / / / P h d [7] h t [9] AF = 5 AF = 7The homotopy class σ { d } [7] is detected by h d [7] in S , which has Adams filtration 5, while itsimage in π ( P ) is detected by h t [9] by Proposition 4.6, which has Adams filtration 7. Therefore,to prove Proposition 4.7, we only need to rule out surviving cycles in the Adams filtration 6, whichalso lie in the kernel of the map P / / / / P in the Adams E ∞ -page. Note that the element h d [7] is not present in Ext ( P ). Proof.
We have the Adams E -page of P and P ∞ in the 46 and 47 stems for s ≤ Ext , ( P ): P h h [6] , h g [2] , h x [8] , h x [9] . Remark 4.5 rules out the last two candidates, since they do not lie in the kernel of the map P / / / / P in the Adams E ∞ -page.In the table for the transfer map in [21], we have that the element h g [2] maps to B . If theimage of the homotopy class σ { d } [7] were detected by h g [2], then we would have a σ -extensionfrom h d to B in π S , which by Lemma 2.2 is equivalent to an η -extension from h g to B in π S . However, the proof of Lemma 4.2.47 of [5] shows the latter is not true.The only candidate left is P h h [6]. To rule it out, we notice there is a long h tower in the 46stem of P ∞ : from h [15] to P h h [4]. In particular, we have h · P h h [6] = P h h [5] , h · P h h [5] = P h h [4] . Since 2 · σ { d } = 0 , the image of the homotopy class σ { d } [7] must have order 2. Therefore, we only need to show theelement P h h [5] is not a boundary. In the following Lemma 4.8, we show that the elements inAdams filtration 4 to 6 of Ext ( P ∞ ) are all permanent cycles. This only leaves the possibility that h h [13] kills P h h [4], but not P h h [5], and hence completes the proof. (cid:3) Lemma 4.8.
The elements in Adams filtration 4 to 6 of the 47-stem of
Ext ( P ∞ ) are all permanentcycles.Proof. There are 7 elements: h h d [1] , h x [9] , h g [2] , h f [6] , h h [2] , g [3] , f [7] . OME EXTENSIONS IN THE ADAMS SPECTRAL SEQUENCE AND THE 51-STEM 11
Table 3.
The Adams E -page of P and P ∞ in the 46 and 47 stems for s ≤ Ext ( P ) Ext ( P ∞ ) s \ t − s
46 47 46 478
P h h [4] • P h h [4] •• •• • P h h [5] • P h h [5] • h t [9] • h t [9] • h x [9] • •• P h h [6] • P h h [6] h h d [1] h g [2] • h g [2] h x [9] h x [8] • h x [8] h x [9]5 h h h [8] • h h h [8] h g [2] • • • h f [6] • • •• • • h h [12] h h [2] • g [3] • f [7] • • • h h [13] • •• h h [14] h h [13]1 h [15]The spheres S , S , S are H F -subcomplexes of P ∞ by the solution of the Hopf invariant oneproblem. Since the elements h h d , g , f are permanent cycles in the Adams spectral sequencefor S , The elements h h d [1] , h g [2] , f [7] are permanent cycles.The element h f [6] = h · f [7] is therefore also a permanent cycle.It is straightforward to show that the elements h g [2] and h h [2] are permanent cycles in theAdams spectral sequence of P . By naturality they are permanent cycles in that of P ∞ .For the element h x [9], one uses the H F -subcomplex of P ∞ which contains cells in dimensions 3,5, 7, 9 to show that it is a permanent cycle. In fact, by comparing the Atiyah-Hirzebruch spectralsequence with the Adams spectral sequence of this 4 cell complex, it follows from the followingrelations in the stable homotopy groups of spheres:0 ∈ η · { h x } , ∈ h ν, η, { h x }i . The homotopy class { h x } [9] survives in the Atiyah-Hirzebruch spectral sequence, and is detectedby h x [9] in its Adams E -page. In particular, h x [9] is a permanent cycle in the Adams spectralsequence of this 4 cell complex, and therefore also a permanent cycle in the Adams spectral sequenceof P ∞ . (cid:3) Now we prove Step 4.
Lemma 4.9.
The element h t [9] maps to N under the transfer map.Proof. We check the two tables in the appendix of [21]. See [21] for more details of the Lambdaalgebra notation we use here. The element N is in Ext , ( S ) = Z /
2. Checking the table for P ∞ , we have that Ext , ( P ∞ ) = ( Z / , generated by (5) 11 12 4 5 3 3 3 , (9) 3 5 7 3 5 7 7 , which means Ext , ( P ∞ ) is generated by P h h [5] and h t [9]. Since P h h [5] is divisible by h in Ext ( P ∞ ), while N is not divisible by h in Ext ( S ), P h h [5] cannot map to N under thetransfer map. By the algebraic Kahn-Priddy theorem [11], the other generator h t [9] has to mapto N . (cid:3) A lemma for extensions in the Atiyah-Hirzebruch spectral sequence
Let α : Y → X and β : Z → Y be homotopy classes of maps between spectra. Suppose that thecomposite αβ = 0. Let Cα and Cβ be the cofiber of α and β respectively.We have cofiber sequences: Y α −→ X i α −→ Cα ∂ α −−→ Σ Y,Z β −→ Y i β −→ Cβ ∂ β −→ Σ Z. Denote by L α β be the set of maps in [Σ Z, Cα ] such that the compositeΣ Z → Cα ∂ α −−→ Σ Y is − Σ β . The indeterminacy of the set L α β is i α · [Σ Z, X ] . Similarly, denote by L β α be the set of maps in [ Cβ, X ] such that the composite Y i β −→ Cβ → X is α . The indeterminacy of the set L β α is [Σ Z, X ] · ∂ β . Lemma 5.1.
The two sets of maps L α β · ∂ β and i α · L β α in [ Cβ, Cα ] are equal.Proof. It is clear that the indeterminacy of the two sets are given by the following composition i α · [Σ Z, X ] · ∂ β . We need to show that they contain one common element. We have the following diagram Y α / / X i α / / Cα ∂ α / / Σ Y − Σ α / / Σ XZ β / / Y i β / / Cβ f O O ✤✤✤✤✤✤ ∂ β / / Σ Z g O O ✤✤✤✤✤✤ − Σ β / / Σ Y Take f ∈ L β α . Since both lines are cofiber sequences, there exists a co-extension g ∈ L α β suchthat the diagram commutes. The commutativity of the middle square gives the claim. (cid:3) OME EXTENSIONS IN THE ADAMS SPECTRAL SEQUENCE AND THE 51-STEM 13
Lemma 5.2.
Let W γ −→ Z β −→ Y α −→ X be a sequence of homotopy classes of maps. Suppose that αβ = 0 and βγ = 0 . Then the two sets ofmaps − L α β · Σ γ and i α · L β α · L β γ in [Σ W, Cα ] are equal.Proof. First, the indeterminacy of the former is i α · [Σ Z, X ] · Σ γ . The indeterminacy of the latter is i α · [Σ Z, X ] · ∂ β · L β γ + i α · L β α · i β · [Σ W, Y ]Note that ∂ β · L β γ = − Σ γ and i α · L β α · i β = i α · α = 0. So the two sets have the same indeterminacy.We have the following diagram: W Σ − L β γ (cid:15) (cid:15) γ / / Z i γ / / Cγ L γ β (cid:15) (cid:15) ∂ γ / / Σ W L β γ (cid:15) (cid:15) − Σ γ / / Σ Z Σ − Cβ Σ − L β α (cid:15) (cid:15) − Σ − ∂ β / / Z Σ − L α β (cid:15) (cid:15) β / / Y i β / / Cβ L β α (cid:15) (cid:15) ∂ β / / Σ Z L α β (cid:15) (cid:15) − Σ β / / Σ Y Σ − X − Σ − i α / / Σ − Cα − Σ − ∂ α / / Y α / / X i α / / Cα ∂ α / / Σ Y By Lemma 5.1, with suitable choices, all the squares commute, then claim follows. In fact, takingany choices of L β γ and L β α , Lemma 5.1 says there exist choices for L γ β and L α β , making thediagrams commute. (cid:3) Now we have the following lemma as a corollary of Lemma 5.2 when the spectra
X, Y, Z, W are all spheres.
Lemma 5.3.
Let α, β and γ be maps between spheres. α : S | α | → S , β : S | α | + | β | → S | α | , γ : S | α | + | β | + | γ | → S | α | + | β | . Then in the Atiyah-Hirzebruch spectral sequence of Cα , we have a γ -extension β [ | α | + 1] · Σ γ = h α, β, γ i [0] Proof.
By definition, the set of classes represented by β [ | α | + 1] in the Atiyah-Hirzebruch spectralsequence is − L α β . On the other hand, by definition, L β α · L β γ is h α, β, γ i , and i α · L β α · L β γ is h α, β, γ i [0]. So the claim follows from Lemma 5.2. (cid:3) Appendix
In this appendix, we use cell diagrams as intuition for the statements of the lemmas in Section5. It is very helpful when thinking of CW spectra. See [1, 20, 22] for example. For simplicity,we restrict to the cases when the spectra
X, Y, Z, W are all spheres. For the definition of celldiagrams, see [1].Let α, β be classes in the stable homotopy groups of spheres such that β · α = 0. We denote thecofiber of α by (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) α (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) We denote the maps i α and ∂ α by (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) α (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) α / / (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) / / (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) and the extension and co-extension maps L α β and L α β by (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) α (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) β / / (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) α (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) β / / (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) Then Lemma 5.1 says the following two sets of maps are equal: (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) β / / (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) β (cid:30) (cid:30) ❃❃❃❃❃❃❃❃ (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) α (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) β (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) α (cid:30) (cid:30) ❃❃❃❃❃❃❃❃ (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) α (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) / / (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) with the same indeterminacy: (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) β / / (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) φ (cid:23) (cid:23) ✳✳✳✳✳✳✳✳✳✳✳✳✳ (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) α (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) / / (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) where φ ∈ π | α | + | β | +1 S could be any class.Suppose further that β · γ = 0. Pre-composing with L β γ , Lemma 5.3 says that the following twosets of maps are equal: (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) γ / / (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) β / / (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) α (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19)(cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) γ / / (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) β (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) α (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) α / / (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) / / (cid:23)(cid:22)(cid:21)(cid:20)(cid:16)(cid:17)(cid:18)(cid:19) OME EXTENSIONS IN THE ADAMS SPECTRAL SEQUENCE AND THE 51-STEM 15
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