Higher chromatic Thom spectra via unstable homotopy theory
aa r X i v : . [ m a t h . A T ] A p r HIGHER CHROMATIC THOM SPECTRA VIA UNSTABLE HOMOTOPYTHEORY
SANATH DEVALAPURKAR
Abstract.
We investigate implications of an old conjecture in unstable homotopy theoryrelated to the Cohen-Moore-Neisendorfer theorem and a conjecture about the E -topologicalHochschild cohomology of certain Thom spectra (denoted A , B , and T ( n )) related to Ravenel’s X ( p n ). We show that these conjectures imply that the orientations MSpin → bo andMString → tmf admit spectrum-level splittings. This is shown by generalizing a theorem ofHopkins and Mahowald, which constructs H F p as a Thom spectrum, to construct BP h n − i ,bo, and tmf as Thom spectra (albeit over T ( n ), A , and B respectively, and not over thesphere). This interpretation of BP h n − i , bo, and tmf offers a new perspective on Woodequivalences of the form bo ∧ Cη ≃ bu: they are related to the existence of certain EHPsequences in unstable homotopy theory. This construction of BP h n − i also provides a dif-ferent lens on the nilpotence theorem. Finally, we prove a C -equivariant analogue of ourconstruction, describing H Z as a Thom spectrum. Contents
1. Introduction 12. Background, and some classical positive and negative results 63. Some Thom spectra 104. Review of some unstable homotopy theory 205. Chromatic Thom spectra 246. Applications 317. C -equivariant analogue of Corollary B 408. Future directions 43References 451. Introduction
Statement of the main results.
One of the goals of this paper is to describe a programto prove the following old conjecture (studied, for instance, in [Lau04, LS19], and discussedinformally in many places, such as [MR09, Section 7]):
Conjecture 1.1.1.
The Ando-Hopkins-Rezk orientation (see [AHR10] ) MString → tmf admitsa spectrum-level splitting. The key idea in our program is to provide a universal property for mapping out of thespectrum tmf. We give a proof which is conditional on an old conjecture from unstable homotopytheory stemming from the Cohen-Moore-Neisendorfer theorem and a conjecture about the E -topological Hochschild cohomology of certain Thom spectra (the latter of which simplifies theproof of the nilpotence theorem from [DHS88]). This universal property exhibits tmf as acertain Thom spectrum, similarly to the Hopkins-Mahowald construction of H Z p and H F p asThom spectra. o illustrate the gist of our argument in a simpler case, recall Thom’s classical result from[Tho54]: the unoriented cobordism spectrum MO is a wedge of suspensions of H F . The simplestway to do so is to show that MO is a H F -module, which in turn can be done by constructingan E -map H F → MO. The construction of such a map is supplied by the following theoremof Hopkins and Mahowald:
Theorem (Hopkins-Mahowald; see [Mah79] and [MRS01, Lemma 3.3]) . Let µ : Ω S → BO denote the real vector bundle over Ω S induced by extending the map S → BO classifying theM¨obius bundle. Then the Thom spectrum of µ is equivalent to H F as an E -algebra. The Thomification of the E -map µ : Ω S → BO produces the desired E -splitting H F → MO.Our argument for Conjecture 1.1.1 takes this approach: we shall show that an old conjecturefrom unstable homotopy theory and a conjecture about the E -topological Hochschild homologyof certain Thom spectra provide a construction of tmf (as well as bo and BP h n i ) as a Thomspectrum, and utilize the resulting universal property of tmf to construct an (unstructured) maptmf → MString.Mahowald was the first to consider the question of constructing spectra like bo and tmf asThom spectra (see [Mah87]). Later work by Rudyak in [Rud98] sharpened Mahowald’s resultsto show that bo and bu cannot appear as the Thom spectrum of a p -complete spherical fibration.Recently, Chatham has shown in [Cha19] that tmf ∧ cannot appear as the Thom spectrum ofa structured 2-complete spherical fibration over a loop space. Our goal is to argue that theseissues are alleviated if we replace “spherical fibrations” with “bundles of R -lines” for certainwell-behaved spectra R .The first hint of tmf being a generalized Thom spectrum comes from a conjecture of Hopkinsand Hahn regarding a construction of the truncated Brown-Peterson spectra BP h n i as Thomspectra. To state this conjecture, we need to recall some definitions. Recall (see [DHS88]) that X ( n ) denotes the Thom spectrum of the map ΩSU( n ) → ΩSU ≃ BU. Upon completion at aprime p , the spectra X ( k ) for p n ≤ k ≤ p n +1 − T ( n ), which in turn filter the gap between the p -completesphere spectrum and BP (in the sense that T (0) = S and T ( ∞ ) = BP). Then: Conjecture 1.1.2 (Hahn, Hopkins) . There is a map f : Ω S | v n | +3 → BGL ( T ( n )) , whichdetects an indecomposable element v n ∈ π | v n | T ( n ) on the bottom cell of the source, whose Thomspectrum is a form of BP h n − i . The primary obstruction to proving that a map f as in Conjecture 1.1.2 exists stems from thefailure of T ( n ) to be an E -ring (due to Lawson; [Law19, Example 1.5.31]). Hahn suggested thatone way to get past the failure of T ( n ) to be an E -ring would be via the following conjecture: Conjecture 1.1.3 (Hahn) . There is an indecomposable element v n ∈ π | v n | T ( n ) which lifts tothe E -topological Hochschild cohomology Z ( X ( p n )) of X ( p n ) . We do not know how to prove this conjecture (and have no opinion on whether or not it istrue). We shall instead show that Conjecture 1.1.2 is implied by the two conjectures alluded toabove. We shall momentarily state these conjectures precisely as Conjectures D and E; let usfirst state our main results.We need to introduce some notation. Let y ( n ) (resp. y Z ( n )) denote the Mahowald-Ravenel-Shick spectrum, constructed as a Thom spectrum over Ω J p n − ( S ) (resp. Ω J p n − ( S ) h i ) intro-duced in [MRS01] to study the telescope conjecture (resp. in [AQ19] as z ( n )). Let A denote the E -quotient S //ν of the sphere spectrum by ν ∈ π ( S ); its mod 2 homology is H ∗ ( A ) ∼ = F [ ζ ].Let B denote the E -ring introduced in [Dev19b, Construction 3.1]. It may be constructed s the Thom spectrum of a vector bundle over an E -space N which sits in a fiber sequenceΩ S → N → Ω S . The mod 2 homology of B is H ∗ ( B ) ∼ = F [ ζ , ζ ].We also need to recall some unstable homotopy theory. In [CMN79a, CMN79b, Nei81], Cohen,Moore, and Neisendorfer constructed a map φ n : Ω S n +1 → S n − whose composite with thedouble suspension S n − → Ω S n +1 is the degree p map. Such a map was also constructed byGray in [Gra89b, Gra88]. In Section 4.1, we introduce the related notion of a charming map (Definition 4.1.1), one example of which is the Cohen-Moore-Neisendorfer map.Our main result is then: Theorem A.
Suppose R is a base spectrum of height n as in the second line of Table 1. Let K n +1 denote the fiber of a charming map Ω S p n +1 +1 → S p n +1 − . Then Conjectures D and Eimply that there is a map K n +1 → BGL ( R ) such that the mod p homology of the Thom spectrum K µn +1 is isomorphic to the mod p homology of the associated designer chromatic spectrum Θ( R ) as a Steenrod comodule .If R is any base spectrum other than B , the Thom spectrum K µn +1 is equivalent to Θ( R ) upon p -completion for every prime p . If Conjecture F is true, then the same is true for B : the Thomspectrum K µn +1 is equivalent to Θ( B ) = tmf upon -completion. Height 0 1 2 n n n
Base spectrum R S ∧ p A B T ( n ) y ( n ) y Z ( n )Designer chromatic spectrum Θ( R ) H Z p bo tmf BP h n i k ( n ) k Z ( n ) Table 1.
To go from a base spectrum “of height n ”, say R , in the second lineto the third, one takes the Thom spectrum of a bundle of R -lines over K n +1 .Na¨ıvely making sense of Theorem A relies on knowing that T ( n ) admits the structure of an E -ring; we shall interpret this phrase as in Warning 3.1.6. In Remark 5.4.7, we sketch howTheorem A relates to the proof of the nilpotence theorem.Although the form of Theorem A does not resemble Conjecture 1.1.2, we show that TheoremA implies the following result. Corollary B.
Conjectures D and E imply Conjecture 1.1.2.
In the case n = 0, Corollary B recovers the Hopkins-Mahowald theorem constructing H F p .Moreover, Corollary B is true unconditionally when n = 0 , R is an E - or E -ring spectrum, let Z ( R ) denote the E -topological Hochschildcohomology of R . Theorem C.
Assume that the composite Z ( B ) → B → MString is an E -map. Then Conjec-tures D, E, and F imply that there is a spectrum-level unital splitting of the Ando-Hopkins-Rezkorientation MString (2) → tmf (2) . In particular, Conjecture 1.1.1 follows (at least after localizing at p = 2; a slight modificationof our arguments should work at any prime). We believe that the assumption that the composite Z ( B ) → B → MString is an E -map is too strong: we believe that it can be removed usingspecial properties of fibers of charming maps, and we will return to this in future work.We stress that these splittings are unstructured; it seems unlikely that they can be refinedto structured splittings. In [Dev19b], we showed (unconditionally) that the Ando-Hopkins-Rezkorientation MString → tmf induces a surjection on homotopy, a result which is clearly impliedby Theorem C. We elected to use the symbol Θ because the first two letters of the English spelling of Θ and of Thom’s nameagree. e remark that the argument used to prove Theorem C shows that if the composite Z ( A ) → A → MSpin is an E -map, then Conjectures D and E imply that there is a spectrum-level unitalsplitting of the Atiyah-Bott-Shapiro orientation MSpin → bo. This splitting was originallyproved unconditionally (i.e., without assuming Conjecture D or Conjecture E) by Anderson-Brown-Peterson in [ABP67] via a calculation with the Adams spectral sequence.1.2. The statements of Conjectures D, E, and F.
We first state Conjecture D. The secondpart of this conjecture is a compilation of several old conjectures in unstable homotopy theoryoriginally made by Cohen-Moore-Neisendorfer, Gray, and Selick in [CMN79a, CMN79b, Nei81,Gra89b, Gra88, Sel77]. The statement we shall give momentarily differs slightly from the state-ments made in the literature: for instance, in Conjecture D(b), we demand a Q -space splitting(Notation 2.2.6), rather than merely a H-space splitting. Conjecture D.
The following statements are true: (a)
The homotopy fiber of any charming map (Definition 4.1.1) is equivalent as a loop spaceto the loop space on an Anick space. (b)
There exists a p -local charming map f : Ω S p n +1 → S p n − whose homotopy fiberadmits a Q -space retraction off of Ω ( S p n /p ) . There are also integrally defined maps Ω S → S and Ω S → S whose composite with the double suspension on S and S respectively is the degree map, whose homotopy fibers K and K (respectively)admit deloopings, and which admit Q -space retractions off Ω ( S / and Ω ( S / (respectively). Next, we turn to Conjecture E. This conjecture is concerned with the E -topological Hochschildcohomology of the Thom spectra X ( p n − ( p ) , A , and B introduced above. Conjecture E.
Let n ≥ be an integer. Let R denote X ( p n +1 − ( p ) , A (in which case n = 1 ), or B (in which case n = 2 ). Then the element σ n ∈ π | v n |− R lifts to the E -topologicalHochschild cohomology Z ( R ) of R , and is p -torsion in π ∗ Z ( R ) if R = X ( p n +1 − ( p ) , and is -torsion in π ∗ Z ( R ) if R = A or B . Finally, we state Conjecture F. It is inspired by [AP76, AL17]. We believe this conjecture isthe most approachable of the conjectures stated here.
Conjecture F.
Suppose X is a spectrum which is bounded below and whose homotopy groupsare finitely generated over Z p . If there is an isomorphism H ∗ ( X ; F p ) ∼ = H ∗ (tmf; F p ) of Steenrodcomodules, then there is a homotopy equivalence X ∧ p → tmf ∧ p of spectra. After proving Theorem A and Theorem C, we explore relationships between the differentspectra appearing on the second line of Table 1 in the remainder of the paper. In particular, weprove analogues of Wood’s equivalence bo ∧ Cη ≃ bu (see also [Mat16]) for these spectra. Weargue that these are related to the existence of certain EHP sequences.Finally, we describe a C -equivariant analogue of Corollary B at n = 1 as Theorem 7.2.1,independently of a C -equivariant analogue of Conjecture D and Conjecture E. This resultconstructs H Z as a Thom spectrum of an equivariant bundle of invertible T (1) R -modules overΩ ρ S ρ +1 , where T (1) R is the free E σ -algebra with a nullhomotopy of the equivariant Hopf map e η ∈ π σ ( S ), and ρ and σ are the regular and sign representations of C , respectively. We believethere is a similar result at odd primes, but we defer discussion of this. We discuss why ourmethods do not work to yield BP h n i R for n ≥ Outline.
Section 2 contains a review some of the theory of Thom spectra from the modernperspective, as well as the proof of the classical Hopkins-Mahowald theorem. The contentreviewed in this section will appear in various guises throughout this project, hence its inclusion. n Section 3, we study certain E -rings; most of them appear as Thom spectra over thesphere. For instance, we recall some facts about Ravenel’s X ( n ) spectra, and then define andprove properties about the E -rings A and B used in the statement of Theorem A. We stateConjecture E, and discuss (Remark 5.4.7) its relation to the nilpotence theorem, in this section.In Section 4, we recall some unstable homotopy theory, such as the Cohen-Moore-Neisendorfermap and the fiber of the double suspension. These concepts do not show up often in stablehomotopy theory, so we hope this section provides useful background to the reader. We stateConjecture D, and then explore properties of Thom spectra of bundles defined over Anick spaces.In Section 5, we state and prove Theorem A and Corollary B, and state several easy conse-quences of Theorem A.In Section 6, we study some applications of Theorem A. For instance, we use it to proveTheorem C, which is concerned with the splitting of certain cobordism spectra. Then, we provethat the Thom spectra in the second line of Table 1 satisfy Wood-like equivalences, and arguethat these equivalences are related to EHP sequences. We also study an approximation to thetopological Hochschild homology of the chromatic Thom spectra of Table 1.In Section 7, we prove an equivariant analogue of Corollary B at height 1. We constructequivariant analogues of X ( n ) and A , and describe why our methods fail to produce an equi-variant analogue of Corollary B at all heights, even granting an analogue of Conjecture D andConjecture E.Finally, in Section 8, we suggest some directions for future research. There are also numerousinteresting questions arising from our work, which we have indicated in the body of the paper.1.4. Conventions.
Unless indicated otherwise, or if it goes against conventional notationalchoices, a Latin letter with a numerical subscript (such as x ) denotes an element of degreegiven by its subscript. If X is a space and R is an E -ring spectrum, then X µ will denote theThom spectrum of some bundle of invertible R -modules determined by a map µ : X → BGL ( R ).We shall often quietly localize or complete at an implicit prime p . Although we have tried tobe careful, all limits and colimits will be homotopy limits and colimits ; we apologize for anyinconvenience this might cause.We shall denote by P k ( p ) the mod p Moore space S k − ∪ p e k with top cell in dimension k .The symbols ζ i and τ i will denote the conjugates of the Milnor generators (commonly writtennowadays as ξ i and τ i , although, as Haynes Miller pointed out to me, our notation for theconjugates was Milnor’s original notation) in degrees 2( p i −
1) and 2 p i − p > i − ζ i ) at p = 2. Unfortunately, we will use A to denote the E -ring in appearing in Table 1,and write A ∗ to denote the dual Steenrod algebra. We hope this does not cause any confusion,since we will always denote the homotopy groups of A by π ∗ A and not A ∗ .If O is an operad, we will simply write O -ring to denote an O -algebra object in spectra. Amap of O -rings respecting the O -algebra structure will often simply be called a O -map. Unlessit is clear that we mean otherwise, all modules over non- E ∞ -algebras will be left modules.Hood Chatham pointed out to me that S h i would be the correct notation for what wedenote by S h i = fib( S → K ( Z , S h i as the preferred notation, so we stick to that in this project.When we write that Theorem A, Corollary B, or Theorem C implies a statement P, we meanthat Conjectures D and Conjecture E (and Conjecture F, if the intended application is to tmf)imply P via Theorem A, Corollary B, or Theorem C.1.5. Acknowledgements.
The genesis of this project took place in conversations with JeremyHahn, who has been a great friend and mentor, and a fabulous resource; I’m glad to be able toacknowledge our numerous discussions, as well as his comments on a draft of this paper. I’mextremely grateful to Mark Behrens and Peter May for working with me over the summer of
019 and for being fantastic advisors, as well as for arranging my stay at UChicago, where partof this work was done. I’d also like to thank Haynes Miller for patiently answering my numerous(often silly) questions over the past few years. Conversations related to the topic of this projectalso took place at MIT and Boulder, and I’d like to thank Araminta Gwynne, Robert Burklund,Hood Chatham, Peter Haine, Mike Hopkins, Tyler Lawson, Andrew Senger, Neil Strickland,and Dylan Wilson for clarifying discussions. Although I never got the chance to meet MarkMahowald, my intellectual debt to him is hopefully evident (simply
Ctrl+F his name in thisdocument!). Finally, I’m glad I had the opportunity to meet other math nerds at the UChicagoREU; I’m in particular calling out Ada, Anshul, Eleanor, and Wyatt — thanks for making mysummer enjoyable.2.
Background, and some classical positive and negative results
Background on Thom spectra.
In this section, we will recall some facts about Thomspectra and their universal properties; the discussion is motivated by [ABG + Definition 2.1.1.
Let A be an E -ring, and let µ : X → BGL ( A ) be a map of spaces. TheThom A -module X µ is defined as the homotopy pushoutΣ ∞ + GL ( A ) / / (cid:15) (cid:15) Σ ∞ + fib( µ ) (cid:15) (cid:15) A / / X µ . Remark 2.1.2.
Let A be an E -ring, and let µ : X → BGL ( A ) be a map of spaces. The Thom A -module X µ is the homotopy colimit of the functor X µ −→ BGL ( A ) → Mod( A ), where we haveabused notation by identifying X with its associated Kan complex. If A is an E - R -algebra,then the R -module underlying X can be identified with the homotopy colimit of the compositefunctor X µ −→ BGL ( A ) → B Aut R ( A ) → Mod( R ) , where we have identified X with its associated Kan complex. The space B Aut R ( A ) can beregarded as the maximal subgroupoid of Mod( R ) spanned by the object A .The following is immediate from the description of the Thom spectrum as a Kan extension: Proposition 2.1.3.
Let R and R ′ be E -rings with an E -ring map R → R ′ exhibiting R ′ asa right R -module. If f : X → BGL ( R ) is a map of spaces, then the Thom spectrum of thecomposite X → BGL ( R ) → BGL ( R ′ ) is the base-change X f ∧ R R ′ of the (left) R -moduleThom spectrum X f . Corollary 2.1.4.
Let R and R ′ be E -rings with an E -ring map R → R ′ exhibiting R ′ as aright R -module. If f : X → BGL ( R ) is a map of spaces such that the the composite X → BGL ( R ) → B GL ( R ′ ) is null, then there is an equivalence X f ∧ R R ′ ≃ R ′ ∧ Σ ∞ + X . Moreover (see e.g. [AB19, Corollary 3.2]):
Proposition 2.1.5.
Let X be a k -fold loop space, and let R be an E k +1 -ring. Then the Thomspectrum of an E k -map X → BGL ( R ) is an E k - R -algebra. We will repeatedly use the following classical result, which is again a consequence of theobservation that Thom spectra are colimits, as well as the fact that total spaces of fibrationsmay be expressed as colimits; see also [Bea17, Theorem 1]. roposition 2.1.6. Let X i −→ Y → Z be a fiber sequence of k -fold loop spaces (where k ≥ ),and let R be an E m -ring for m ≥ k + 1 . Suppose that µ : Y → BGL ( R ) is a map of k -foldloop spaces. Then, there is a k -fold loop map φ : Z → BGL ( X µ ◦ i ) whose Thom spectrumis equivalent to Y µ as E k − -rings. Concisely, if arrows are labeled by their associated Thomspectra, then there is a diagram X i / / X µ ◦ i $ $ ❍❍❍❍❍❍❍❍❍ Y / / µ Y µ (cid:15) (cid:15) Z φ Y µ = Z φ (cid:15) (cid:15) BGL ( R ) / / B GL ( X µ ◦ i ) . The argument to prove Proposition 2.1.6 also goes through with slight modifications when k = 0, and shows: Proposition.
Let X i −→ Y → Z be a fiber sequence of spaces with Z connected, and let R be an E m -ring for m ≥ . Suppose that µ : Y → BGL ( R ) is a map of Kan complexes. Then, there isa map φ : Z → B Aut R ( X µ ◦ i ) whose Thom spectrum Z φ is equivalent to Y µ as an R -module. We will abusively refer to this result in the sequel also as Proposition 2.1.6.The following is a slight generalization of one of the main results of [AB19]:
Theorem 2.1.7.
Let R be an E k +1 -ring for k ≥ , and let α : Y → BGL ( R ) be a map froma pointed space Y . For any ≤ m ≤ k , let e α : Ω m Σ m Y → BGL ( R ) denote the extension of α . Then the Thom spectrum (Ω m Σ m Y ) e α is the free E m - R -algebra A for which the composite Y → BGL ( R ) → BGL ( A ) is null. More precisely, if A is any E m - R -algebra, then Map
Alg E mR ((Ω m Σ m Y ) e α , A ) ≃ ( Map ∗ ( Y, Ω ∞ A ) if α : Y → BGL ( R ) → BGL ( A ) is null ∅ else . Remark 2.1.8.
Say Y = S n +1 , so α detects an element α ∈ π n R . Corollary 2.1.7 suggestsinterpreting the Thom spectrum (Ω m S m + n +1 ) e α as an E m -quotient; to signify this, we willdenote it by R// E m α . If m = 1, then we will simply denote it by R//α , while if m = 0, then the E m -quotient is simply the ordinary quotient R/α .2.2.
The Hopkins-Mahowald theorem.
The primary motivation for this project is the fol-lowing miracle (see [Mah79] for p = 2 and [MRS01, Lemma 3.3] for p > Theorem 2.2.1 (Hopkins-Mahowald) . Let S ∧ p be the p -completion of the sphere at a prime p ,and let f : S → BGL ( S ∧ p ) detect the element − p ∈ π BGL ( S ∧ p ) ≃ Z × p . Let µ : Ω S → BGL ( S ∧ p ) denote the E -map extending f ; then there is a p -complete equivalence (Ω S ) µ → H F p of E -ring spectra. It is not too hard to deduce the following result from Theorem 2.2.1:
Corollary 2.2.2.
Let S h i denote the -connected cover of S . Then the Thom spectrum ofthe composite Ω S h i → Ω S µ −→ BGL ( S ∧ p ) is equivalent to H Z p as an E -ring. Remark 2.2.3.
Theorem 2.2.1 implies a restrictive version of the nilpotence theorem: if R isan E -ring spectrum, and x ∈ π ∗ R is a simple p -torsion element which has trivial H F p -Hurewiczimage, then x is nilpotent. This is explained in [MNN15, Proposition 4.19]. Indeed, to showthat x is nilpotent, it suffices to show that the localization R [1 /x ] is contractible. Since px = 0,the localization R [1 /x ] is an E -ring in which p = 0, so the universal property of Theorem 2.1.7implies that there is an E -map H F p → R [1 /x ]. It follows that the unit R → R [1 /x ] factors hrough the Hurewicz map R → R ∧ H F p . In particular, the multiplication by x map on R [1 /x ]factors as the indicated dotted map:Σ | x | R (cid:15) (cid:15) x / / R / / (cid:15) (cid:15) R [1 /x ] . H F p ∧ Σ | x | R x / / R ∧ H F p sssss However, the bottom map is null (because x has trivial H F p -Hurewicz image), so x must be nullin π ∗ R [1 /x ]. This is possible if and only if R [1 /x ] is contractible, as desired. See Proposition5.4.1 for the analogous connection between Corollary 2.2.2 and nilpotence.Since an argument similar to the proof of Theorem 2.2.1 will be necessary later, we will recalla proof of this theorem. The key non-formal input is the following result of Steinberger’s from[BMMS86]: Theorem 2.2.4 (Steinberger) . Let ζ i denote the conjugate to the Milnor generators ξ i of thedual Steenrod algebra, and similarly for τ i at odd primes. Then Q p i ζ i = ζ i +1 , Q p j τ j = τ j +1 for i, j + 1 > .Proof of Theorem 2.2.1. By Corollary 2.1.7, the Thom spectrum (Ω S ) µ is the free E -ringwith a nullhomotopy of p . Since H F p is an E -ring with a nullhomotopy of p , we obtain an E -map (Ω S ) µ → H F p . To prove that this map is a p -complete equivalence, it suffices toprove that it induces an isomorphism on mod p homology.The mod p homology of (Ω S ) µ can be calculated directly via the Thom isomorphism H F p ∧ (Ω S ) µ ≃ H F p ∧ Σ ∞ + Ω S . Note that this is not an equivalence as H F p ∧ H F p -comodules: theThom twisting is highly nontrivial.For simplicity, we will now specialize to the case p = 2, although the same proof works atodd primes. The homology of Ω S is classical: it is a polynomial ring generated by applying E -Dyer-Lashof operations to a single generator x in degree 1. Theorem 2.2.4 implies thatthe same is true for the mod 2 Steenrod algebra: it, too, is a polynomial ring generated byapplying E -Dyer-Lashof operations to the single generator ζ = ξ in degree 1. Since the map(Ω S ) µ → H F is an E -ring map, it preserves E -Dyer-Lashof operations on mod p homology.By the above discussion, it suffices to show that the generator x ∈ H ∗ (Ω S ) µ ∼ = H ∗ (Ω S ) indegree 1 is mapped to ζ ∈ H ∗ H F .To prove this, note that x is the image of the generator in degree 1 in homology underthe double suspension S → Ω S , and that ζ is the image of the generator in degree 1 inhomology under the canonical map S /p → H F p . It therefore suffices to show that the Thomspectrum of the spherical fibration S → BGL ( S ∧ p ) detecting 1 − p is simply S /p . This is aneasy exercise. (cid:3) Remark 2.2.5.
When p = 2, one does not need to p -complete in Theorem 2.2.1: the map S → BGL ( S ∧ ) factors as S → BO → B GL ( S ), where the first map detects the M¨obiusbundle over S , and the second map is the J -homomorphism. Notation 2.2.6.
Let Q denote the (operadic nerve of the) cup-1 operad from [Law19, Example1.3.6]: this is the operad whose n th space is empty unless n = 2, in which case it is S with theantipodal action of Σ . We will need to slightly modify the definition of Q when localized at anodd prime p : in this case, it will denote the operad whose n th space is a point if n < p , emptyif n > p , and when n = p is the ordered configuration space Conf p ( R ) with the permutationaction of Σ p . Any homotopy commutative ring admits the structure of a Q -algebra at p = 2, ut at other primes it is slightly stronger to be a Q -algebra than to be a homotopy commutativering. If k ≥
2, any E k -algebra structure on a spectrum restricts to a Q -algebra structure. Remark 2.2.7.
As stated in [Law19, Proposition 1.5.29], the operation Q already exists inthe mod 2 homology of any Q -ring R , where Q is the cup-1 operad from Notation 2.2.6 — theentire E -structure is not necessary. With our modification of Q at odd primes as in Remark2.2.6, this is also true at odd primes. Remark 2.2.8.
We will again momentarily specialize to p = 2 for convenience. Steinberger’scalculation in Theorem 2.2.4 can be rephrased as stating that Q ζ i = ζ i +1 , where Q is the lower-indexed Dyer-Lashof operation. As in Remark 2.2.7, the operation Q already exists inthe mod p homology of any Q -ring R . Since homotopy commutative rings are Q -algebras inspectra, this observation can be used to prove results of W¨urgler [Wur86] and Pazhitnov-Rudyak[PR84]. Remark 2.2.9.
The argument with Dyer-Lashof operations and Theorem 2.2.4 used in theproof of Theorem 2.2.1 will be referred to as the
Dyer-Lashof hopping argument . It will be usedagain (in the same manner) in the proof of Theorem A.
Remark 2.2.10.
Theorem 2.2.1 is equivalent to Steinberger’s calculation (Theorem 2.2.4), aswell as to B¨okstedt’s calculation of THH( F p ) (as a ring spectrum, and not just the calculationof its homotopy).2.3. No-go theorems for higher chromatic heights.
In light of Theorem 2.2.1 and Corollary2.2.2, it is natural to wonder if appropriate higher chromatic analogues of H F p and H Z , such asBP h n i , bo, or tmf can be realized as Thom spectra of spherical fibrations. The answer is knownto be negative (see [Mah87, Rud98, Cha19]) in many cases: Theorem 2.3.1 (Mahowald, Rudyak, Chatham) . There is no space X with a spherical fibration µ : X → BGL ( S ) (even after completion) such that X µ is equivalent to BP h i or bo . Moreover,there is no -local loop space X ′ with a spherical fibration determined by a H-map µ : X ′ → BGL ( S ∧ ) such that X ′ µ is equivalent to tmf ∧ . The proofs rely on calculations in the unstable homotopy groups of spheres.
Remark 2.3.2.
Although not written down anywhere, a slight modification of the argumentused by Mahowald to show that bu is not the Thom spectrum of a spherical fibration over aloop space classified by a H-map can be used to show that BP h i at p = 2 (i.e., tmf (3)) is notthe Thom spectrum of a spherical fibration over a loop space classified by a H-map. We do notknow a proof that BP h n i is not the Thom spectrum of a spherical fibration over a loop spaceclassified by a H-map for all n ≥ Remark 2.3.3.
A lesser-known no-go result due to Priddy appears in [Lew78, Chapter 2.11],where it is shown that BP cannot appear as the Thom spectrum of a double loop map Ω X → BGL ( S ). In fact, the argument shows that the same result is true with BP replaced by BP h n i for n ≥
1; we had independently come up with this argument for BP h i before learning aboutPriddy’s argument. Since Lewis’ thesis is not particularly easy to acquire, we give a sketch ofPriddy’s argument. By the Thom isomorphism and the calculation (see [LN14, Theorem 4.3]):H ∗ (BP h n − i ; F p ) ∼ = ( F [ ζ , · · · , ζ n − , ζ n , ζ n +1 , · · · ] p = 2 F p [ ζ , ζ , · · · ] ⊗ Λ F p ( τ n , τ n +1 , · · · ) p > , we find that the mod p homology of Ω X would be isomorphic as an algebra to a polynomialring on infinitely many generators, possibly tensored with an exterior algebra on infinitely many enerators. The Eilenberg-Moore spectral sequence then implies that the mod p cohomology of X is given by H ∗ ( X ; F p ) ∼ = ( F [ b , · · · , b n , c n +1 , · · · ] p = 2 F p [ b , b , · · · ] ⊗ Λ F p ( c n +1 , · · · ) p > , where | b i | = 2 p i and | c i | = 2 p i − + 1. If p is odd, then since | b | = 2 p , we have P ( b ) = b p .However, Liulevicius’ formula for P in terms of secondary cohomology operations from [Liu62]involves cohomology operations of odd degrees (which kills b because everything is concentratedin even degrees in the relevant range) and a cohomology operation of degree 4( p −
1) (which alsokills b since 4( p −
1) + 2 p i is never a sum of numbers of the form 2 p k when p > b p = 0, which is a contradiction. A similar calculation works at p = 2, using Adams’ studyof secondary mod 2 cohomology operations in [Ada60].Our primary goal in this project is to argue that the issues in Theorem 2.3.1 are alleviated ifwe replace BGL ( S ) with the delooping of the space of units of an appropriate replacement of S . In the next section, we will construct these replacements of S .3. Some Thom spectra
In this section, we introduce certain E -rings; most of them appear as Thom spectra overthe sphere. The following table summarizes the spectra introduced in this section and givesreferences to their locations in the text. The spectra A and B were introduced in [Dev19b].Thom spectrum Definition “Height” BP-homology T ( n ) Theorem 3.1.5 n Theorem 3.1.5 y ( n ) and y Z ( n ) Definition 3.2.1 n Proposition 3.2.2 A Definition 3.2.7 1 Proposition 3.2.11 B Definition 3.2.16 2 Proposition 3.2.18
Table 2.
Certain Thom spectra and their homologies.3.1.
Ravenel’s X ( n ) spectra. The proof of the nilpotence theorem in [DHS88, HS98] cruciallyrelied upon certain Thom spectra arising from Bott periodicity; these spectra first appeared inRavenel’s seminal paper [Rav84].
Definition 3.1.1.
Let X ( n ) denote the Thom spectrum of the E -map ΩSU( n ) ⊆ BU J −→ BGL ( S ), where the first map arises from Bott periodicity. Example 3.1.2.
The E -ring X (1) is the sphere spectrum, while X ( ∞ ) is MU. Since themap ΩSU( n ) → BU is an equivalence in dimensions ≤ n −
2, the same is true for the map X ( n ) → MU; the first dimension in which X ( n ) has an element in its homotopy which is notdetected by MU is 2 n − Remark 3.1.3.
The E -structure on X ( n ) does not extend to an E -structure (see [Law19,Example 1.5.31]). If X ( n ) admits such an E -structure, then the induced map H ∗ ( X ( n )) → H ∗ (H F p ) on mod p homology would commute with E -Dyer-Lashof operations. However, weknow that the image of H ∗ ( X ( n )) in H ∗ (H F p ) is F p [ ζ , · · · , ζ n ]; since Steinberger’s calculation(Theorem 2.2.4) implies that Q ( ζ i ) = ζ i +1 via the relation Q ( x ) = Q ( x ) , we find that theimage of H ∗ ( X ( n )) in H ∗ (H F p ) cannot be closed under the E -Dyer-Lashof operation Q . Remark 3.1.4.
The proof of the nilpotence theorem shows that each of the X ( n ) detectsnilpotence. However, it is known (see [Rav84]) that h X ( n ) i > h X ( n + 1) i . fter localizing at a prime p , the spectrum MU splits as a wedge of suspensions of BP; thissplitting comes from the Quillen idempotent on MU. The same is true of the X ( n ) spectra, asexplained in [Rav86, Section 6.5]: a multiplicative map X ( n ) ( p ) → X ( n ) ( p ) is determined bya polynomial f ( x ) = P ≤ i ≤ n − a i x i +1 , with a = 1 and a i ∈ π i ( X ( n ) ( p ) ). One can use thisto define a truncated form of the Quillen idempotent ǫ n on X ( n ) ( p ) (see [Hop84]), and therebyobtain a summand of X ( n ) ( p ) . We summarize the necessary results in the following theorem. Theorem 3.1.5.
Let n be such that p n ≤ k ≤ p n +1 − . Then X ( k ) ( p ) splits as a wedge ofsuspensions of the spectrum T ( n ) = ǫ p n · X ( p n ) ( p ) . • The map T ( n ) → BP is an equivalence in dimensions ≤ | v n +1 | − , so there is anindecomposable element v i ∈ π ∗ T ( n ) which maps to an indecomposable element in π ∗ BP for ≤ i ≤ n . • This map induces the inclusion BP ∗ T ( n ) = BP ∗ [ t , · · · , t n ] ⊆ BP ∗ (BP) on BP -homology,and the inclusions F [ ζ , · · · , ζ n ] ⊆ F [ ζ , ζ , · · · ] and F p [ ζ , · · · , ζ n ] ⊆ F [ ζ , ζ , · · · ] onmod and mod p homology. • T ( n ) is a homotopy associative and Q -algebra spectrum. Warning 3.1.6.
We have not proved that T ( n ) admits the structure of an E -ring (even thoughit is true; this is unpublished work of Beardsley-Hahn-Lawson). We shall nonetheless abusivelyinterpret the phrase “Thom spectrum X µ of a map µ : X → BGL ( T ( n ))” where µ arises via amap X µ −→ BGL ( X ( p n +1 − X µ ∧ X ( p n +1 − T ( n ). Note that thenotation BGL ( T ( n )) is meaningless unless T ( n ) admits an E -structure.It is believed that T ( n ) in fact admits more structure: Conjecture 3.1.7.
The Q -ring structure on T ( n ) extends to an E -ring structure. Remark 3.1.8.
This is true at p = 2 and n = 1. Indeed, in this case T (1) = X (2) is the Thomspectrum of the bundle given by the 2-fold loop map Ω S = Ω BSU(2) → BU, induced by theinclusion BSU(2) → B U = BSU.
Remark 3.1.9.
Conjecture 3.1.7 is true at p = 2 and n = 2. The Stiefel manifold V ( H ) sitsin a fiber sequence S → V ( H ) → S . There is an equivalence V ( H ) ≃ Sp(2), so Ω V ( H ) admits the structure of a double loopspace. There is an E -map µ : Ω V ( H ) → BU, given by taking double loops of the compositeBSp(2) → BSU(4) → BSU ≃ B U . The map µ admits a description as the left vertical map in the following map of fiber sequences:Ω V ( H ) / / µ (cid:15) (cid:15) Ω S / / (cid:15) (cid:15) S (cid:15) (cid:15) BU / / ∗ / / B U . Here, the map S → B U detects the generator of π (BU) (which maps to η ∈ π (BGL ( S ))under the J-homomorphism). The Thom spectrum Ω V ( H ) µ is equivalent to T (2), and itfollows that T (2) admits the structure of an E -ring. We do not know whether T ( n ) is theThom spectrum of a p -complete spherical fibration over some space for n ≥ X ( n + 1) as an X ( n )-algebra (see also [Bea18]): ησ = ην ν Figure 1. Cη ∧ Cν shown horizontally, with 0-cell on the left. The element σ is shown. Construction 3.1.10.
There is a fiber sequenceΩSU( n ) → ΩSU( n + 1) → Ω S n +1 . According to Proposition 2.1.6, the spectrum X ( n + 1) is the Thom spectrum of an E -map Ω S n +1 → BGL (ΩSU( n )) µ = BGL ( X ( n )). This E -map is the extension of a map S n → BGL ( X ( n )) which detects an element χ n ∈ π n − X ( n ). This element is equivalentlydetermined by the map Σ ∞ + Ω S n +1 → X ( n ) given by the Thomification of the nullhomotopiccomposite Ω S n +1 → ΩSU( n ) → ΩSU( n + 1) → ΩSU ≃ BU , where the first two maps form a fiber sequence. By Proposition 2.1.6, X ( n + 1) is the free E - X ( n )-algebra with a nullhomotopy of χ n . Remark 3.1.11.
Another construction of the map χ n ∈ π n − X ( n ) from Construction 3.1.10is as follows. There is a map i : C P n − → ΩSU( n ) given by sending a line ℓ ⊆ C n to the loop S → SU( n ) = Aut( C n , h , i ) defined as follows: θ ∈ S is sent to the (appropriate rescaling ofthe) unitary transformation of C n sending a vector to its rotation around the line ℓ by the angle θ . The map i Thomifies to a stable map Σ − C P n → X ( n ). The map χ n is then the composite S n − → Σ − C P n → X ( n ) , where the first map is the desuspension of the generalized Hopf map S n +1 → C P n whichattaches the top cell of C P n +1 . The fact that this map is indeed χ n follows immediately fromthe commutativity of the following diagram: S n − / / (cid:15) (cid:15) C P n − / / (cid:15) (cid:15) C P n (cid:15) (cid:15) Ω S n +1 / / ΩSU( n ) / / ΩSU( n + 1) , where the top row is a cofiber sequence, and the bottom row is a fiber sequence.An easy consequence of the observation in Construction 3.1.10 is the following lemma. Lemma 3.1.12.
Let σ n ∈ π | v n +1 |− T ( n ) denote the element χ p n +1 − . Then the Thom spectrumof the composite Ω S | v n +1 | +1 → BGL ( X ( p n +1 − → BGL ( T ( n )) is equivalent to T ( n + 1) . Example 3.1.13.
The element σ ∈ π | v |− T (0) = π p − S ( p ) is α . Example 3.1.14.
Let us specialize to p = 2. Theorem 3.1.5 implies that H ∗ T ( n ) ∼ = F [ ζ , · · · , ζ n ].Using this, one can observe that the 6-skeleton of T (1) is the smash product Cη ∧ Cν , and so σ ∈ π ( Cη ∧ Cν ). This element can be described very explicitly: the cell structure of Cη ∧ Cν is shown in Figure 1, and the element σ shown corresponds to the map defined by the relation ην = 0. Example 3.1.15.
The element σ n in the Adams-Novikov spectral sequence for T ( n ) is repre-sented by the element [ t n +1 ] in the cobar complex. calculation with the Adams-Novikov spectral sequence (as in [Rav02, Theorem 3.17], where σ n − is denoted by α (ˆ v /p )) proves the following: Lemma 3.1.16.
The group π | v n |− X ( p n − is isomorphic to Z /p . In particular, the element σ n − = χ p n − ∈ π | v n |− X ( p n −
1) is p -torsion, and the following isa consequence of Example 3.1.15: Proposition 3.1.17.
The Toda bracket h p, σ n − , X ( p n ) i in π | v n | X ( p n ) contains an indecom-posable v n . Corollary 3.1.18.
Let ≤ m ≤ n . There is a choice of indecomposable v m ∈ π | v m | X ( p n ) whichlifts to π | v m | +2 (Σ ∞ C P p m ) along the map Σ − C P p m → X ( p n ) . Similarly, σ n − ∈ π | v n |− X ( p n − lifts to π | v n | +1 ( C P | v n | / ) along the map Σ − C P | v n | / → X ( p n − .Proof. By Remark 3.1.11, the map σ n − : S | v n |− → X ( p n −
1) is given by the compositeof the generalized Hopf map S | v n |− → Σ − C P p n − with the map Σ − C P p n − → X ( p n − S | v n | +1 → C P p n − , and so σ n − lifts to an element of the unstable homotopy group π | v n | +1 ( C P | v n | / ).We now describe a construction of v m ∈ π | v m | X ( p n ). There is a cofiber sequence C P p m → S p m → Σ C P p m − , and the attaching map (i.e., the generalized Hopf map) Σ σ m − : S p m → Σ C P p m − is stably p -torsion. Therefore, p times this map admits a lift to C P p m , and hence defines an element of π p m ( C P p m ). This maps to an element of π p m − ( X ( p n )) under the map Σ − C P p m → X ( p n ).The construction makes it clear that this element lies in the Toda bracket h p, σ m − , X ( p n ) i , sowe conclude by Proposition 3.1.17. (cid:3) Remark 3.1.19.
The conclusion of Corollary 3.1.18 for the element v is well-known: namely, v lifts to π p ( C P p ). Namely, the p th power of the inclusion β : C P = S ֒ → C P ∞ definesan element of π p (Σ ∞ C P ∞ ), and the map β p : S p → C P ∞ factors through the 2 p -skeleton C P p of C P ∞ . The resulting element of π p ( C P p ) maps to v ∈ π p − X ( p ) under the mapΣ − C P p → X ( p ).3.2. Related Thom spectra.
We now introduce several Thom spectra related to the E -rings T ( n ) described in the previous section; some of these were introduced in [Dev19b]. (Relationshipsto T ( n ) will be further discussed in Section 6.2.) For the reader’s convenience, we have included atable of the spectra introduced below with internal references to their definitions at the beginningof this section.The following Thom spectrum was introduced in [MRS01]. Definition 3.2.1.
Let y ( n ) denote the Thom spectrum of the compositeΩ J p n − ( S ) → Ω S − p −−→ BGL ( S ∧ p ) . If J p n − ( S ) h i denotes the 2-connected cover of J p n − ( S ), then let y Z ( n ) denote the Thomspectrum of the composite Ω J p n − ( S ) h i → Ω S − p −−→ BGL ( S ∧ p ) . The following proposition is stated for y ( n ) in [MRS01, Equation 2.8]. Proposition 3.2.2. As BP ∗ BP -comodules, we have BP ∗ ( y ( n )) ∼ = BP ∗ /I n [ t , · · · , t n ] , BP ∗ ( y Z ( n )) ∼ = BP ∗ / ( v , · · · , v n − )[ t , · · · , t n ] , where I n denotes the invariant ideal ( p, v , · · · , v n − ) . n immediate corollary is the following. Corollary 3.2.3. As A ∗ -comodules, we have H ∗ ( y ( n ); F p ) ∼ = ( F [ ζ , ζ , · · · , ζ n ] p = 2 F p [ ζ , ζ , · · · , ζ n ] ⊗ E ( τ , · · · , τ n − ) p ≥ , and H ∗ ( y Z ( n ); F p ) ∼ = ( F [ ζ , ζ , · · · , ζ n ] p = 2 F p [ ζ , ζ , · · · , ζ n ] ⊗ E ( τ , · · · , τ n − ) p ≥ . We will now relate y ( n ) and y Z ( n ) to T ( n ). Construction 3.2.4.
Let m ≤ n , and let I n be the ideal generated by p, v , · · · , v m − , where the v i are some choices of indecomposables in π | v i | ( T ( n )) which form a regular sequence. Inductivelydefine T ( n ) /I m as the cofiber of the map T ( n ) /I m − v m ∧ −−−→ T ( n ) ∧ T ( n ) /I m − → T ( n ) /I m − . The BP-homology of T ( n ) /I m is BP ∗ /I m [ t , · · · , t n ]. The spectrum T ( n ) / ( v , · · · , v m − ) isdefined similarly. Proposition 3.2.5.
Let p > . There is an equivalence between T ( n ) /I n (resp. T ( n ) / ( v , · · · , v n − ) )and the spectrum y ( n ) (resp. y Z ( n ) ) of Definition 3.2.1.Proof. We will prove the result for y ( n ); the analogous proof works for y Z ( n ). By [Gra89a], thespace Ω J p n − ( S ) is homotopy commutative (since p > J p n − ( S ) → Ω S is a H-map, so y ( n ) is a homotopy commutative E -ring spectrum. It is known (see [Rav86,Section 6.5]) that homotopy commutative maps T ( n ) → y ( n ) are equivalent to partial complexorientations of y ( n ), i.e., factorizations S / / $ $ ❏❏❏❏❏❏❏❏❏❏❏ Σ − C P p n − γ n (cid:15) (cid:15) ✤✤✤ y ( n ) . Such a γ n indeed exists by obstruction theory: suppose k < p n −
1, and we have a mapΣ − C P k → y ( n ). Since there is a cofiber sequence S k − → Σ − C P k → Σ − C P k +1 of spectra, the obstruction to extending along Σ − C P k +1 is an element of π k − y ( n ). However,the homotopy of y ( n ) is concentrated in even degrees in the appropriate range, so a choice of γ n does indeed exist. Moreover, this choice can be made such that they fit into a compatiblefamily in the sense that there is a commutative diagramΣ − C P p n − / / γ n (cid:15) (cid:15) Σ − C P p n +1 − γ n +1 (cid:15) (cid:15) y ( n ) / / y ( n + 1) . The formal group law over H F p has infinite height; this forces the elements p, v , · · · , v n − (defined for the “( p n − π ∗ y ( n )) to vanish in the homotopy of y ( n ). It follows thatthe orientation T ( n ) → y ( n ) constructed above factors through the quotient T ( n ) /I n . Theinduced map T ( n ) /I n → y ( n ) can be seen to be an isomorphism on homology (via, for instance,Definition 3.2.1 and Construction 3.2.4). (cid:3) emark 3.2.6. Since y ( n ) has a v n -self-map, we can form the spectrum y ( n ) /v n ; its mod p homology is H ∗ ( y ( n ) /v n ; F p ) ∼ = ( F [ ζ , · · · , ζ n ] ⊗ Λ F ( ζ n +1 ) p = 2 F p [ ζ , · · · , ζ n ] ⊗ Λ F p ( τ , · · · , τ n − , τ n ) p ≥ . It is in fact possible to give a construction of y (1) /v as a spherical Thom spectrum. We willwork at p = 2 for convenience. Define Q to be the fiber of the map 2 η : S → S . There is amap of fiber sequences Q / / (cid:15) (cid:15) S η / / (cid:15) (cid:15) S (cid:15) (cid:15) BGL ( S ) / / ∗ / / B GL ( S ) . By [DM81, Theorem 3.7], the Thom spectrum of the leftmost map is y (1) /v .We end this section by recalling the definition of two Thom spectra which, unlike y ( n ) and y Z ( n ), are not indexed by integers (we will see that they are only defined at “heights 1 and 2”).These were both introduced in [Dev19b]. Definition 3.2.7.
Let S → BSpin denote the generator of π BSpin ∼ = Z , and let Ω S → BSpindenote the extension of this map, which classifies a real vector bundle of virtual dimension zeroover Ω S . Let A denote the Thom spectrum of this bundle. Remark 3.2.8.
As mentioned in the introduction, the spectrum A has been intensely studied byMahowald and his coauthors in (for instance) [Mah79, DM81, Mah81b, Mah81a, Mah82, MU77],where it is often denoted X . Remark 3.2.9.
The map Ω S → BSpin is one of E -spaces, so the Thom spectrum A admitsthe structure of an E -ring with an E -map A → MSpin.
Remark 3.2.10.
There are multiple equivalent ways to characterize this Thom spectrum. Forinstance, the J -homomorphism BSpin → BGL ( S ) sends the generator of π BSpin to ν ∈ π BGL ( S ) ∼ = π S . The universal property of Thom spectra in Theorem 2.1.7 shows that A isthe free E -ring S //ν with a nullhomotopy of ν . Note that A is defined integrally, and not just p -locally for some prime p .The following result is [Dev19b, Proposition 2.6]; it is proved there at p = 2, but the argumentclearly works for p = 3 too. Proposition 3.2.11.
There is an isomorphism BP ∗ ( A ) ∼ = BP ∗ [ y ] , where | y | = 4 . There is amap A ( p ) → BP . Under the induced map on BP -homology, y maps to t mod decomposables at p = 2 , and to t mod decomposables at p = 3 . Remark 3.2.12.
For instance, when p = 2, we have BP ∗ ( A ) ∼ = BP ∗ [ t + v t ].An immediate corollary is the following. Corollary 3.2.13. As A ∗ -comodules, we have H ∗ ( A ; F p ) ∼ = F [ ζ ] p = 2 F [ ζ ] p = 3 F p [ x ] p ≥ , where x is a polynomial generator in degree . ν νη σ σ Figure 2. A at the prime 2 shown horizontally, with 0-cell onthe left. The element σ is shown. Example 3.2.14.
Let us work at p = 2 for convenience. Example 3.1.14 showed that σ is theelement in π ( Cη ∧ Cν ) given by the lift of η to the 4-cell (which is attached to the bottom cellby ν ) via a nullhomotopy of ην . In particular, σ already lives in π ( Cν ), and as such definesan element of S //ν = A (by viewing Cν as the 4-skeleton of A ); note that, by construction, thiselement is 2-torsion. There is a canonical map A → T (1), and the image of σ ∈ π ( A ) underthis map is its namesake in π ( T (1)). See Figure 2. Remark 3.2.15.
The element σ ∈ π ( A (2) ) defined in Example 3.2.14 in fact lifts to an elementof π ( A ), because the relation ην = 0 is true integrally, and not just 2-locally. An alternateconstruction of this map is the following. The Hopf map η : S → S (which lives in the stablerange) defines a map S → S → Ω S whose composite to BSpin is null (since π (BSpin) = 0).Upon Thomification of the composite S → Ω S → BSpin, one therefore gets a map S → A whose composite with A → MSpin is null. The map S → A is the element σ ∈ π ( A ).Finally, we have: Definition 3.2.16.
Let B N be the space defined by the homotopy pullbackB N / / (cid:15) (cid:15) S f (cid:15) (cid:15) BO(9) / / BO(10) , where the map f : S → BO(10) detects an element of π O(10) ∼ = Z /
12. There is a fibersequence S → BO(9) → BO(10) , and the image of f under the boundary map in the long exact sequence of homotopy detects2 ν ∈ π ( S ) ∼ = Z /
24. In particular, there is a fiber sequence S → B N → S . If N is defined to be ΩB N , then there is a fiber sequence N → Ω S → S . Define a map N → BString via the map of fiber sequences N / / (cid:15) (cid:15) Ω S / / (cid:15) (cid:15) S (cid:15) (cid:15) BString / / ∗ / / B String , where the map S → B String detects a generator of π BString. The map N → BString is infact one of E -spaces. Let B denote the Thom spectrum of the induced bundle over N . σ η σ Figure 3.
Steenrod module structure of the 20-skeleton of B ; the bottom cell(in dimension 0) is on the left; straight lines are Sq , and curved lines correspondto Sq and Sq , in order of increasing length. The bottom two attaching mapsof B are labeled. The map σ is shown.We introduced the spectrum B and studied its Adams-Novikov spectral sequence in [Dev19b].The Steenrod module structure of the 20-skeleton of B is shown in [Dev19b, Figure 1], and isreproduced here as Figure 3. As mentioned in the introduction, the spectrum B has been brieflystudied under the name X in [HM02]. Remark 3.2.17.
As with A , there are multiple different ways to characterize B . There is afiber sequence Ω S → N → Ω S , and the map Ω S → N → BString is an extension of the map S → BString detecting agenerator. Under the J -homomorphism BString → BGL ( S ), this generator maps to σ ∈ π BGL ( S ) ∼ = π S , so the Thom spectrum of the bundle over Ω S determined by the mapΩ S → BString is the free E -ring S //σ with a nullhomotopy of ν . Proposition 2.1.6 now impliesthat N is the Thom spectrum of a map Ω S → BGL ( S //σ ). While a direct definition of thismap is not obvious, we note that the restriction to the bottom cell S of the source detects anelement e ν of π BGL ( S //σ ) ∼ = π S //σ . This in turn factors through the 11-skeleton of S //σ ,which is the same as the 8-skeleton of S //σ (namely, Cσ ). This element is precisely a lift of themap ν : S → S to Cσ determined by a nullhomotopy of σν in π ∗ S . It is detected by h inthe Adams spectral sequence.The following result is [Dev19b, Proposition 3.2]; it is proved there at p = 2, but the argumentclearly works for p ≥ Proposition 3.2.18.
The BP ∗ -algebra BP ∗ ( B ) is isomorphic to a polynomial ring BP ∗ [ b , y ] ,where | b | = 8 and | y | = 12 . There is a map B ( p ) → BP . On BP ∗ -homology, the elements b and y map to t and t mod decomposables at p = 2 , and y maps to t mod decomposables at p = 3 . An immediate corollary is the following.
Corollary 3.2.19. As A ∗ -comodules, we have H ∗ ( B ; F p ) ∼ = F [ ζ , ζ ] p = 2 F [ ζ , b ] p = 3 F [ ζ , x ] p = 5 F p [ x , x ] p ≥ , where x and x are polynomial generators in degree and , and b is an element in degree . Example 3.2.20.
For simplicity, let us work at p = 2. There is a canonical ring map B → T (2),and the element σ ∈ π T (2) lifts to B . We can be explicit about this: the 12-skeleton of B is shown in Figure 3, and σ is the element of π ( B ) existing thanks to the relation ην = 0 nd the fact that the Toda bracket h η, ν, σ i contains 0. This also shows that σ ∈ π ( B ) is2-torsion. Remark 3.2.21.
The element σ ∈ π ( B (2) ) defined in Example 3.2.20 in fact lifts to anelement of π ( B ), because the relations νσ = 0, ην = 0, and 0 ∈ h η, ν, σ i are all true integrally,and not just 2-locally. An alternate construction of this map S → B is the following. TheHopf map η : S → S (which lives in the stable range) defines a map S → S → Ω S .Moreover, the composite S → Ω S → S is null, since it detects an element of π ( S ) = 0;choosing a nullhomotopy of this composite defines a lift S → N . (In fact, this comes from amap S → B N .) The composite S → N → BString is null (since π (BString) = 0). UponThomification, we obtain a map S → B whose composite with B → MString is null; the map S → B is the element σ ∈ π ( B ).The following theorem packages some information contained in this section. Theorem 3.2.22.
Let R denote any of the spectra in Table 2, and let n denote its “height”. If R = T ( n ) , y ( n ) , or y Z ( n ) , then there is a map T ( n ) → R , and if R = A (resp. B ), then there isa map from T (1) (resp. T (2) ) to R . In the first three cases, there is an element σ n ∈ π | v n +1 |− R coming from σ n ∈ π | v n +1 |− T ( n ) , and in the cases R = A and B , there are elements σ ∈ π ( A ) and σ ∈ π ( B ) mapping to the corresponding elements in T (1) (2) and T (2) (2) , respectively.Moreover, σ n is p -torsion in π ∗ R ; similarly, σ and σ are -torsion in π ∗ A (2) and π ∗ B (2) .Proof. The existence statement for T ( n ) is contained in Theorem 3.1.5, while the torsion state-ment is the content of Lemma 3.1.16. The claims for y ( n ) and y Z ( n ) now follow from Proposition3.2.5. The existence and torsion statements for A and B are contained in Examples 3.2.14 and3.2.20. (cid:3) The elements in Theorem 3.2.22 can in fact be extended to infinite families; this is discussedin Section 5.4.3.3.
Centers of Thom spectra.
In this section, we review some of the theory of E k -centersand state Conjecture E. We begin with the following important result, and refer to [Fra13] and[Lur16, Section 5.5.4] for proofs. Theorem 3.3.1.
Let C be a symmetric monoidal presentable ∞ -category, and let A be an E k -algebra in C . Then the category of E k - A -modules is equivalent to the category of left modulesover the factorization homology U ( A ) = R S k − × R A (known as the enveloping algebra of A ),which is an E -algebra in C . Definition 3.3.2.
The E k +1 -center Z ( A ) of an E k -algebra A in C is the ( E k +1 -)Hochschildcohomology End U ( A ) ( A ), where A is regarded as a left module over its enveloping algebra viaTheorem 3.3.1. Remark 3.3.3.
We are using slightly different terminology than the one used in [Lur16, Sec-tion 5.3]: our E k +1 -center is his E k -center. In other words, Lurie’s terminology expresses thestructure on the input, while our terminology expresses the structure on the output.The following proposition summarizes some results from [Fra13] and [Lur16, Section 5.3]. Proposition 3.3.4.
The E k +1 -center Z ( A ) of an E k -algebra A in a symmetric monoidal pre-sentable ∞ -category C exists, and satisfies the following properties: (a) Z ( A ) is the universal E k -algebra of C which fits into a commutative diagram A / / ❍❍❍❍❍❍❍❍❍❍❍ ❍❍❍❍❍❍❍❍❍❍❍ A ⊗ Z ( A ) (cid:15) (cid:15) A n Alg E k ( C ) . (b) The E k -algebra Z ( A ) of C defined via this universal property in fact admits the structureof an E k +1 -algebra in C . (c) There is a fiber sequence GL ( Z ( A )) → GL ( A ) → Ω k − End
Alg E k (Sp) ( A ) of k -fold loop spaces. In the sequel, we will need a more general notion:
Definition 3.3.5.
Let m ≥
1. The E k + m -center Z k + m ( A ) of an E k -algebra A in a presentablesymmetric monoidal ∞ -category C with all limits is defined inductively as the E k + m -center ofthe E k + m − -center Z k + m − ( A ). In other words, it is the universal E k + m -algebra of C which fitsinto a commutative diagram Z k + m − ( A ) / / ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ ❙❙❙❙❙❙❙❙❙❙❙❙❙❙ Z k + m − ( A ) ⊗ Z k + m ( A ) (cid:15) (cid:15) Z k + m − ( A )in Alg E k + m − ( C ).Proposition 3.3.4 gives: Corollary 3.3.6.
Let m ≥ . The E k + m − -algebra Z k + m ( A ) associated to an E k -algebra object A of C exists, and in fact admits the structure of an E k + m -algebra in C . We can now finally state Conjecture E:
Conjecture E.
Let n ≥ be an integer. Let R denote X ( p n +1 − ( p ) , A (in which case n = 1 ),or B (in which case n = 2 ). Then the element σ n ∈ π | σ n | R lifts to the E -center Z ( R ) of R ,and is p -torsion in π ∗ Z ( R ) if R = X ( p n +1 − ( p ) , and is -torsion in π ∗ Z ( R ) if R = A or B . Remark 3.3.7. If R is A or B , then Z ( R ) is the E -center of the E -center of R . This is arather unwieldy object, so it would be quite useful to show that the E -structure on A or B admits an extension to an E -structure; we do not know if such extensions exist. Since neitherΩ S nor N admit the structure of a double loop space, such an E -structure would not arisefrom their structure as Thom spectra. In any case, if such extensions do exist, then Z ( R ) inConjecture E should be interpreted as the E -center of the E -ring R . Remark 3.3.8.
In the introduction, we stated Conjecture 1.1.3, which instead asked aboutwhether v n ∈ π | v n | X ( p n ) lifts to π ∗ Z ( X ( p n )). It is natural to ask about the connection betweenConjecture E and Conjecture 1.1.3. Proposition 3.1.17 implies that if Z ( X ( p n )) admitted an X ( p n − X ( p n − X ( p n − → X ( p n ), and σ n − ∈ π | v n |− X ( p n −
1) was killed by the map X ( p n − → Z ( X ( p n )), then Conjecture Eimplies Conjecture 1.1.3. However, we do not believe that either of these statements are true. Remark 3.3.9.
One of the main results of [Kla18] implies that the E -center of X ( n ) is X ( n ) h SU( n ) , where SU( n ) acts on X ( n ) by a Thomification of the conjugation action on ΩSU( n ). Remark 3.3.10.
Note that the conjugation action of SU( n ) on X ( n ) can be described veryexplicitly, via a concrete model for ΩSU( n ). As explained in [PS86, Zhu17], if G is a reductivelinear algebraic group over C , the loop space Ω G ( C ) of its complex points , viewed as a complex More precisely, the complex points of a maximal compact — but this distinction is irrelevant homotopically,because G ( C ) deformation retracts onto a maximal compact. ie group, is equivalent to the homogeneous space G ( C (( t ))) /G ( C [[ t ]]); this is also commonlystudied as the complex points of the affine Grassmannian Gr G of G . The conjugation actionof G ( C ) on Ω G ( C ) arises by restricting the descent (to G ( C (( t ))) /G ( C [[ t ]])) of the translationaction by G ( C [[ t ]]) on G ( C (( t ))) to the subgroup G ( C ) ⊆ G ( C [[ t ]]). Setting G = SL n givesa description of the conjugation action of SU( n ) on ΩSU( n ). In light of its connections togeometric representation theory, we believe that there may be an algebro-geometric approachto proving that χ n is SU( n )-trivial in X ( n ) and in ΩSU( n ). Example 3.3.11.
The element χ ∈ π X (2) is central. To see this, note that α ∈ π ∗ R (where R is an E k -ring) is in the E k +1 -center of R if and only if α is in the E k +1 -center of R ( p ) for allprimes p ≥
0. It therefore suffices to show that χ is central after p -localizing for all p . First,note that χ is torsion, so it is nullhomotopic (and therefore central) after rationalization. Next,if p >
2, then X (2) ( p ) splits as a wedge of suspensions of spheres. If χ is detected in π ofa sphere living in dimension 3, then it could not be torsion, so it must be detected in π of asphere living in dimension 3 − k for some 0 ≤ k ≤
2. If k = 1 or 2, then π ( S − k ) is either π ( S ) or π ( S ), but both of these groups vanish for p >
2. Therefore, χ must be detected in π of the sphere in dimension 0, i.e., in π X (1). This group vanishes for p >
3, and when p = 3,it is isomorphic to Z / α ). Since X (1) = S is an E ∞ -ring, we conclude that χ is central in X (2) ( p ) for all p > p = 2, we know the cell structure of X (2) in the bottom few dimensions (see Example3.1.14; note that σ is not χ ). In dimensions ≤
3, it is equivalent to Cη , so π X (2) ∼ = π Cη .However, it is easy to see that π Cη = 0, so χ is nullhomotopic at p = 2 (and is thereforevacuously central). We conclude from the above discussion that χ is indeed central in X (2).4. Review of some unstable homotopy theory
Charming and Gray maps.
A major milestone in unstable homotopy theory was Cohen-Moore-Neisendorfer’s result on the p -exponent of unstable homotopy groups of spheres from[CMN79a, CMN79b, Nei81]. They defined for all p > k ≥ φ n : Ω S n +1 → S n − (the integer k is assumed implicit) such that the composite of φ n with the double suspension E : S n − → Ω S n +1 is homotopic to the p k -th power map. By induction on n , they concludedvia a result of Selick’s (see [Sel77]) that p n kills the p -primary component of the homotopy of S n +1 . Such maps will be important in the rest of this paper, so we will isolate their desiredproperties in the definition of a charming map , inspired by [ST19]. (Our choice of terminologyis non-standard, and admittedly horrible, but it does not seem like the literature has chosen anynaming convention for the sort of maps we desire.) Definition 4.1.1. A p -local map f : Ω S np +1 → S np − is called a Gray map if the compositeof f with the double suspension E is the degree p map, and the compositeΩ S n +1 Ω H −−→ Ω S np +1 f −→ S np − is nullhomotopic. Moreover, a p -local map f : Ω S np +1 → S np − is called a charming map ifthe composite of f with the double suspension E is the degree p map, the fiber of f admits thestructure of a Q -space, and if there is a space BK which sits in a fiber sequence S np − → BK → Ω S np +1 such that the boundary map Ω S np +1 → S np − is homotopic to f . Remark 4.1.2. If f is a charming map, then the fiber of f is a loop space. Indeed, fib( f ) ≃ Ω BK . xample 4.1.3. Let f denote the Cohen-Moore-Neisendorfer map with k = 1. Anick proved(see [Ani93, GT10]) that the fiber of f admits a delooping, i.e., there is a space T np +1 ( p ) (nowknown as an Anick space ) which sits in a fiber sequence S np − → T np +1 ( p ) → Ω S np +1 . It follows that f is a charming map. Remark 4.1.4.
We claim that T p +1 ( p ) = Ω S h i , where S h i is the 3-connected cover of S .To prove this, we will construct a p -local fiber sequence S p − → Ω S h i → Ω S p +1 . This fiber sequence was originally constructed by Toda in [Tod62]. To construct this fibersequence, we first note that there is a p -local fiber sequence S p − → J p − ( S ) → C P ∞ , where the first map is the factorization of α : S p − → Ω S through the 2( p − S , and the second map is the composite J p − ( S ) → Ω S → C P ∞ . This fiber sequence issimply an odd-primary version of the Hopf fibration S → S → C P ∞ ; the identification of thefiber of the map J p − ( S ) → C P ∞ is a simple exercise with the Serre spectral sequence. Next,we have the EHP sequence J p − ( S ) → Ω S → Ω S p +1 . Since Ω S h i is the fiber of the map Ω S → C P ∞ , the desired fiber sequence is obtained bytaking vertical fibers in the following map of fiber sequences: J p − ( S ) / / (cid:15) (cid:15) Ω S / / (cid:15) (cid:15) Ω S p +1 (cid:15) (cid:15) C P ∞ C P ∞ / / ∗ . Example 4.1.5.
Let W n denote the fiber of the double suspension S n − → Ω S n +1 . Grayproved in [Gra89b, Gra88] that W n admits a delooping BW n , and that after p -localization, thereis a fiber sequence BW n → Ω S np +1 f −→ S np − for some map f . As suggested by the naming convention, f is a Gray map.As proved in [ST19], Gray maps satisfy an important rigidity property: Proposition 4.1.6 (Selick-Theriault) . The fiber of any Gray map admits a H-space structure,and is H-equivalent to BW n . Remark 4.1.7.
It has been conjectured by Cohen-Moore-Neisendorfer and Gray in the paperscited above that there is an equivalence BW n ≃ Ω T np +1 ( p ), and that Ω T np +1 ( p ) retracts offof Ω P np +1 ( p ) as a H-space, where P k ( p ) is the mod p Moore space S k − ∪ p e k with top cell indimension k . For our purposes, we shall require something slightly stronger: namely, the retrac-tion should be one of Q -spaces. The first part of this conjecture would follow from Proposition4.1.6 if the Cohen-Moore-Neisendorfer map were a Gray map. In [Ame19], it is shown that theexistence of p -primary elements of Kervaire invariant one would imply equivalences of the form BW p n − ≃ Ω T p n +1 ( p ).Motivated by Remark 4.1.7 and Proposition 4.1.6, we state the following conjecture; it isslightly weaker than the conjecture mentioned in Remark 4.1.7, and is an amalgamation of slightmodifications of conjectures of Cohen, Moore, Neisendorfer, Gray, and Mahowald in unstablehomotopy theory, as well as an analogue of Proposition 4.1.6. (For instance, we strengthenhaving a H-space retraction to having a Q -space retraction). onjecture D. The following statements are true: (a)
The homotopy fiber of any charming map is equivalent as a loop space to the loop spaceon an Anick space. (b)
There exists a p -local charming map f : Ω S p n +1 → S p n − whose homotopy fiberadmits a Q -space retraction off of Ω P p n +1 ( p ) . There are also integrally defined maps Ω S → S and Ω S → S whose composite with the double suspension on S and S respectively is the degree map, whose homotopy fibers K and K (respectively)admit deloopings, and which admits a Q -space retraction off Ω P (2) and Ω P (2) (respectively). Remark 4.1.8.
Conjecture D is already not known when n = 1. In this case, it asserts thatΩ S h i retracts off of Ω P p +1 ( p ). A consequence of Selick’s theorem that Ω S h i retractsoff of Ω S p +1 { p } for p odd is that Ω S h i retracts off of Ω P p +2 ( p ). In [Coh87, Observation9.2], the question of whether Ω S h i retracts off of Ω P p +1 ( p ) was shown to be equivalentto the question of whether there is a map Σ Ω S h i → P p +1 ( p ) which is onto in homology.Some recent results regarding Conjecture D for n = 1 can be found in [BT13].It follows that a retraction of Ω S h i off Ω P p +1 ( p ) will be compatible with the canonicalmap Ω S h i → Ω S in the following manner. The p -torsion element α ∈ π p ( S ) defines amap P p − ( p ) → Ω S , which extends to an E -map Ω P p +1 ( p ) → Ω S . We will abusivelydenote this extension by α . The resulting compositeΩ S h i → Ω P p +1 ( p ) α −→ Ω S is homotopic to the canonical map Ω S h i → Ω S .The element α ∈ π p − ( S ( p ) ) defines a map S p − → BGL ( S ( p ) ), and since it is p -torsion,admits an extension to a map P p − ( p ) → BGL ( S ( p ) ). (This extension is in fact unique, because π p − (BGL ( S ( p ) )) ∼ = π p − ( S ( p ) ) vanishes.) Since BGL ( S ( p ) ) is an infinite loop space, this mapfurther extends to a map Ω P p +1 ( p ) → BGL ( S ( p ) ). The discussion in the previous paragraphimplies that if Conjecture D is true for n = 1, then the map µ : Ω S h i → BGL ( S ( p ) ) fromCorollary 2.2.2 is homotopic to the compositeΩ S h i → Ω P p +1 ( p ) → BGL ( S ( p ) ) . Fibers of charming maps.
We shall need the following proposition.
Proposition 4.2.1.
Let f : Ω S p n +1 → S p n − be a charming map. Then there are isomor-phisms of coalgebras: H ∗ (fib( f ); F p ) ∼ = ( F p [ x n +1 − ] ⊗ N k> F p [ x n + k − ] p = 2 N k> F p [ y p n + k − ] ⊗ N j> Λ F p [ x p n + j − ] p > . Proof.
This is an easy consequence of the Serre spectral sequence coupled with the well-knowncoalgebra isomorphismsH ∗ (Ω S n +1 ; F p ) ∼ = (N k> F p [ x k n − ] p = 2 N k> F p [ y np k − ] ⊗ N j ≥ Λ F p [ x np j − ] p > , where these classes are generated by the one in dimension 2 n − (cid:3) Remark 4.2.2.
The Anick spaces T np +1 ( p ) from Example 4.1.3 sit in fiber sequences S np − → T np +1 ( p ) → Ω S np +1 , nd are homotopy commutative H-spaces. A Serre spectral sequence calculation gives an iden-tification of coalgebras H ∗ ( T np +1 ( p ); F p ) ∼ = F p [ a np ] ⊗ Λ F p [ b np − ] , with β ( a np ) = b np − , where β is the Bockstein homomorphism. An argument with the barspectral sequence recovers the result of Proposition 4.2.1 in this particular case. Remark 4.2.3.
The Serre spectral sequence calculating the integral homology of the delooping BK of the fiber of a charming map f : Ω S np +1 → S np − only has a differential on the E np − -page, and givesH ∗ ( BK ; Z ) ∼ = Z [ x k +1) np − k − | k ≥ / ( x i x j , x i , kx k +1) np − k − ) , where x k +1) np − k − lives in degree 2( k + 1) np − k −
1. Explicitly:H i ( BK ; Z ) ∼ = Z i = 0 Z /k if i = 2( k + 1) np − k −
10 else . We conclude this section by investigating Thom spectra of bundles defined over fibers ofcharming maps. Let R be a p -local E -ring, and let µ : K → BGL ( R ) denote a map fromthe fiber K of a charming map f : Ω S np +1 → S np − . There is a fiber sequence Ω S np − → K → Ω S np +1 of loop spaces, so we obtain a map Ω S np − → BGL ( R ). Such a map gives anelement α ∈ π np − R via the effect on the bottom cell S np − .Theorem 2.1.7 implies that the Thom spectrum of the map Ω S np − → BGL ( R ) should bethought of as the E -quotient R//α , although this may not make sense if R is not at least E .However, in many cases (such as the ones we are considering here), the Thom R -module R//α isin fact an E -ring such that the map R → R//α is an E -map. By Proposition 2.1.6, there is aninduced map φ : Ω S np +1 → BGL ( R//α ) whose Thom spectrum is equivalent as an E -ringto K µ . We would like to determine the element of π ∗ R//α detected by the restriction to thebottom cell S np − of the source of φ . First, we note: Lemma 4.2.4.
The element α ∈ π np − R is p -torsion.Proof. Since f is a charming map, the composite S np − → Ω S np − f −→ S np − is the degree p map. Therefore, the element pα ∈ π np − R is detected by the composite S np − → Ω S np − → Ω S np − f −−→ Ω S np − → K µ −→ BGL ( R ) . But there is a fiber sequence Ω S np − f −→ S np − → BK by the definition of a charming map,so the composite detecting pα is null, as desired. (cid:3) There is now a square S np − /p / / (cid:15) (cid:15) S np − (cid:15) (cid:15) K / / α (cid:15) (cid:15) Ω S np +1 (cid:15) (cid:15) BGL ( R ) / / B GL ( R//α ) , Technically, this is bad terminology: there are multiple possibilities for the map φ , and each gives rise to amap S np − → BGL ( R//α ). The elements in π np − ( R//α ) determined in this way need not agree, but theyare the same modulo the indeterminacy of the Toda bracket h p, α, R//α i . nd the following result is a consequence of the lemma and the definition of Toda brackets: Lemma 4.2.5.
The element in π np − ( R//α ) detected by the vertical map S np − → BGL ( R//α ) lives in the Toda bracket h p, α, R//α i . The upshot of this discussion is the following:
Proposition 4.2.6.
Let R be a p -local E -ring, and let µ : K → BGL ( R ) denote a map fromthe fiber K of a charming map f : Ω S np +1 → S np − , providing an element α ∈ π np − R .Assume that the Thom spectrum R//α of the map Ω S np − → BGL ( R ) is an E - R -algebra.Then there is an element v ∈ h p, α, R//α i such that K µ is equivalent to the Thom spectrum ofthe map Ω S np +1 v −→ BGL ( R//α ) . Remark 4.2.7.
Let R be an E -ring, and let α ∈ π d R . Then α defines a map S d +1 → BGL ( R ),and it is natural to ask when α extends along S d +1 → Ω S d +2 , or at least along S d +1 → J k ( S d +1 )for some k . This is automatic if R is an E -ring, but not necessarily so if R is only an E -ring.Recall that there is a cofiber sequence S ( k +1)( d +1) − → J k ( S d +1 ) → J k +1 ( S d +1 ) , where the first map is the ( k + 1)-fold iterated Whitehead product [ ι d +1 , [ · · · , [ ι d +1 , ι d +1 ]] , · · · ].In particular, the map S d +1 → BGL ( R ) extends along the map S d +1 → J k ( S d +1 ) if and only ifthere are compatible nullhomotopies of the n -fold iterated Whitehead products [ α, [ · · · , [ α, α ]] , · · · ] ∈ π ∗ BGL ( R ) for n ≤ k . These amount to properties of Toda brackets in the homotopy of R . Wenote, for instance, that the Whitehead bracket [ α, α ] ∈ π d +1 BGL ( R ) ∼ = π d R is the element2 α ; therefore, the map S d +1 → BGL ( R ) extends to J ( S d +1 ) if and only if 2 α = 0. Remark 4.2.8.
Let R be a p -local E -ring, and let α ∈ π d ( R ) with d even. Then α definesan element α ∈ π d +2 B GL ( R ). The p -fold iterated Whitehead product [ α, · · · , [ α, α ] , · · · ] ∈ π p ( d +2) − ( p − B GL ( R ) ∼ = π pd +( p − R is given by p ! Q ( α ) modulo decomposables.This is in fact true more generally. Let R be an E n -ring, and suppose α ∈ π d ( R ). Let i < n , so α defines an element α ∈ π d + i B i GL ( R ). The p -fold iterated Whitehead product[ α, · · · , [ α, α ] , · · · ] ∈ π p ( d + i ) − ( p − B i GL ( R ) ∼ = π pd +( i − p − R is given by p ! Q i − ( α ) modulodecomposables. We shall expand on this in forthcoming work, but we note here that this is aconsequence of the definition of Whitehead products and properties of (1-fold) loop spaces ofspheres. 5. Chromatic Thom spectra
Statement of the theorem.
To state the main theorem of this section, we set somenotation. Fix an integer n ≥
1, and work in the p -complete stable category. For each Thomspectrum R of height n − σ n − : S | σ n − | → BGL ( R ) denotea map detecting σ n − ∈ π | σ n − | ( R ) (which exists by Theorem 3.2.22). Let K n denote the fiberof a p -local charming map Ω S p n +1 → S p n − satisfying the hypotheses of Conjecture D, andlet K (resp. K ) denote the fiber of an integrally defined charming map Ω S → S (resp.Ω S → S ) satisfying the hypotheses of Conjecture D.Then: Theorem A.
Let R be a height n − spectrum as in the second line of Table 1. Then ConjecturesD and E imply that there is a map K n → BGL ( R ) such that the mod p homology of the Thomspectrum K µn is isomorphic to the mod p homology of the associated designer chromatic spectrum Θ( R ) as a Steenrod comodule.If R is any base spectrum other than B , the Thom spectrum K µn is equivalent to Θ( R ) upon p -completion for every prime p . If Conjecture F is true, then the same is true for B : the Thomspectrum K µn is equivalent to Θ( B ) = tmf upon -completion. eight 0 1 2 n n n Base spectrum S ∧ p A B T ( n ) y ( n ) y Z ( n )Designer chromatic Thom spectrum H Z p bo tmf BP h n i k ( n ) k Z ( n ) Table 3.
A reprint of Table 1, used in the statement of Theorem A.We emphasize again that na¨ıvely making sense of Theorem A relies on knowing that T ( n )admits the structure of an E -ring; we shall interpret this phrase as in Warning 3.1.6. Remark 5.1.1.
Theorem A is proved independently of the nilpotence theorem. (In fact, it iseven independent of Quillen’s identification of π ∗ MU with the Lazard ring, provided one regardsthe existence of designer chromatic spectra as being independent of Quillen’s identification.)We shall elaborate on the connection between Theorem A and the nilpotence theorem in futurework; a sketch is provided in Remark 5.4.7.
Remark 5.1.2.
Theorem A is true unconditionally when n = 1, since that case is simplyCorollary 2.2.2. Remark 5.1.3.
Note that Table 2 implies that the homology of each of the Thom spectra inTable 1 are given by the Q -Margolis homology of their associated designed chromatic spectra.In particular, the map R → Θ( R ) is a rational equivalence.Before we proceed with the proof of Theorem A, we observe some consequences. Corollary B.
Conjecture D and Conjecture E imply Conjecture 1.1.2.Proof.
This follows from Theorem A, Proposition 4.2.6, and Proposition 3.1.17. (cid:3)
Remark 5.1.4.
Corollary B is true unconditionally when n = 1, since Theorem A is trueunconditionally in that case by Remark 5.1.2. See also Remark 4.1.4. Remark 5.1.5.
We can attempt to apply Theorem A for R = A in conjunction with Proposition4.2.6. Theorem A states that Conjecture D and Conjecture E imply that there is a map K → BGL ( A ) whose Thom spectrum is equivalent to bo. There is a fiber sequenceΩ S → K → Ω S , so we obtain a map µ : Ω S → K → BGL ( A ). The proof of Theorem A shows that thebottom cell S of the source detects σ ∈ π ( A ). A slight variation of the argument used toestablish Proposition 4.2.6 supplies a map Ω S → B Aut((Ω S ) µ ) whose Thom spectrum is bo.The spectrum (Ω S ) µ has mod 2 homology F [ ζ , ζ ]. However, unlike A , it does not naturallyarise an E -Thom spectrum over the sphere spectrum; this makes it unamenable to study viatechniques of unstable homotopy.More precisely, (Ω S ) µ is not the Thom spectrum of an E -map X → BGL ( S ) from a loopspace X which sits in a fiber sequence Ω S → X → Ω S of loop spaces. Indeed, B X would be a S -bundle over S , which by [Mah87, Lemma 4] impliesthat X is then equivalent as a loop space to Ω S × Ω S . The resulting E -map Ω S → BGL ( S )is specified by an element of π ( S ) ∼ = 0, so (Ω S ) µ must then be equivalent as an E -ring to A ∧ Σ ∞ + Ω S . In particular, σ ∈ π ( A ) would map nontrivially to (Ω S ) µ , which is a contradiction.The proof of Theorem A will also show: Corollary 5.1.6.
Let R be a height n − spectrum as in the second line of Table 1, and assumeConjecture F if R = B . Let M be an E - R -algebra. Conjecture D and Conjecture E imply thatif: (a) the composite Z ( R ) → R → M is an E -algebra map, b) the element σ n − in π ∗ M is nullhomotopic, (c) and the bracket h p, σ n − , M i contains zero,then there is a unital map Θ( R ) → M . The proof of Theorem A.
This section is devoted to giving a proof of Theorem A,dependent on Conjecture D and Conjecture E. The proof of Theorem A will be broken downinto multiple steps. The result for y ( n ) and y Z ( n ) follow from the result for T ( n ) by Proposition3.2.5, so we shall restrict ourselves to the cases of R being T ( n ), A , and B .Fix n ≥
1. If R is A or B , we will restrict to p = 2, and let K and K denote the integrallydefined spaces from Conjecture D. By Remarks 3.2.15 and 3.2.21, the elements σ ∈ π ( A ) and σ ∈ π ( B ) are defined integrally. We will write σ n − to generically denote this element, andwill write it as living in degree | σ n − | . We shall also write R to denote X ( p n −
1) and not T ( n );this will be so that we can apply Conjecture D. We apologize for the inconvenience, but hopethat this is worth circumventing the task of having to read through essentially the same proofsfor these slightly different cases. Step 1.
We begin by constructing a map µ : K n → BGL ( R ) as required by the statement ofTheorem A; the construction in the case n = 1 follows Remark 4.1.8. By Conjecture D, thespace K n splits off Ω P | σ n − | +4 ( p ) (if R = T ( n ), then | σ n − | + 4 = | v n | + 3). We are thereforereduced to constructing a map Ω P | σ n − | +4 ( p ) → BGL ( R ). Theorem 3.2.22 shows that theelement σ n − ∈ π ∗ R is p -torsion, so the map S | σ n − | +1 → BGL ( R ) detecting σ n − extends toa map(5.1) S | σ n − | +1 /p = P | σ n − | +2 ( p ) → BGL ( R ) . Since Ω P | σ n − | +4 ( p ) ≃ Ω Σ P | σ n − | +2 ( p ), we would obtain an extension e µ of this map throughΩ P | σ n − | +4 ( p ) if R admits an E -structure.Unfortunately, this is not true; but this is where Conjecture E comes in: it says that theelement σ n − ∈ π | σ n − | R lifts to the E -center Z ( R ), where it has the same torsion order as in R . (Here, we are abusively writing Z ( T ( n − E -center of X ( p n − ( p ) .) Thelifting of σ n − to π | σ n − | Z ( R ) provided by Conjecture E gives a factorization of the map from(5.1) as S | σ n − | +1 /p = P | σ n − | +2 ( p ) → BGL ( Z ( R )) → BGL ( R ) . Since Z ( R ) is an E -ring, BGL ( Z ( R )) admits the structure of an E -space. In particular, themap P | σ n − | +2 ( p ) → BGL ( Z ( R )) factors through Ω P | σ n − | +4 ( p ), as desired. We let e µ denotethe resulting composite e µ : Ω P | σ n − | +4 ( p ) → BGL ( Z ( R )) → BGL ( R ) . Step 2.
Theorem A asserts that there is an identification between the Thom spectrum of theinduced map µ : K n → BGL ( R ) and the associated designer chromatic spectrum Θ( R ) viaTable 1. We shall identify the Steenrod comodule structure on the mod p homology of K µn , andshow that it agrees with the mod p homology of Θ( R ).In Table 4, we have recorded the mod p homology of the designer chromatic spectra in Table1 (see [LN14, Theorem 4.3] for BP h n − i ). It follows from Proposition 4.2.1 that there is anisomorphismH ∗ ( K µn ) ∼ = ( H ∗ ( R ) ⊗ F [ x n +1 − ] ⊗ N k> F [ x n + k − ] p = 2H ∗ ( R ) ⊗ N k> F p [ y p n + k − ] ⊗ N j> Λ F p [ x p n + j − ] p > . Combining this isomorphism with Theorem 3.1.5, Proposition 3.2.3, Proposition 3.2.13, andProposition 3.2.19, we find that there is an abstract equivalence between the mod p homologyof K µn and the mod p homology of Θ( R ). esigner chromatic spectrum Mod p homologyBP h n − i p = 2 F [ ζ , · · · , ζ n − , ζ n , ζ n +1 , · · · ] p > F p [ ζ , ζ , · · · ] ⊗ Λ F p ( τ n , τ n +1 , · · · ) k ( n − p = 2 F [ ζ , · · · , ζ n − , ζ n , ζ n +1 , · · · ] p > F p [ ζ , ζ , · · · ] ⊗ Λ F p ( τ , · · · , τ n − , τ n , τ n +1 , · · · ) k Z ( n − p = 2 F [ ζ , ζ , · · · , ζ n − , ζ n , ζ n +1 , · · · ] p > F p [ ζ , ζ , · · · ] ⊗ Λ F p ( τ , · · · , τ n − , τ n , τ n +1 , · · · )bo p = 2 F [ ζ , ζ , ζ , · · · ] p > F p [ x ] /v ⊗ F p [ ζ , ζ , · · · ] ⊗ Λ F p ( τ , τ , · · · )tmf p = 2 F [ ζ , ζ , ζ , ζ , · · · ] p = 3 Λ F ( b ) ⊗ F [ ζ , ζ , · · · ] ⊗ Λ F ( τ , τ , · · · ) p ≥ F p [ c , c ] / ( v , v ) ⊗ F p [ ζ , ζ , · · · ] ⊗ Λ F p ( τ , τ , · · · ) Table 4.
The mod p homology of designer chromatic spectra.We shall now work at p = 2 for the remainder of the proof; the same argument goesthrough with slight modifications at odd primes. We now identify the Steenrod comodulestructure on H ∗ ( K µn ). Recall that e µ is the map Ω P | σ n − | +4 ( p ) → BGL ( R ) from Step 1.By construction, there is a map K µn → Ω P | σ n − | +4 ( p ) e µ . The map Φ factors through a map e Φ : Ω P | σ n − | +4 ( p ) e µ → Θ( R ). The Thom spectrum Ω P | σ n − | +4 ( p ) e µ admits the structure of a Q -ring. Indeed, it is the smash product Ω P | σ n − | +4 ( p ) φ ∧ Z ( R ) R , where φ : Ω P | σ n − | +4 ( p ) → BGL ( Z ( R )); it therefore suffices to observe that the Thom spectrum Ω P | σ n − | +4 ( p ) φ admitsthe structure of an E ⊗ Q -ring. This is a consequence of the fact that φ is a double loop,and hence an E ⊗ Q -algebra, map. Moreover, the image of H ∗ ( K µn ) in H ∗ (Ω P | σ n − | +4 ( p ) e µ ) isgenerated under the single Dyer-Lashof operation (arising from the cup-1 operad; see Remark2.2.8) by the indecomposables in the image of the map H ∗ ( R ) → H ∗ (Ω P | σ n − | +4 ( p ) e µ ).The Postnikov truncation map Ω P | σ n − | +4 ( p ) e µ → H π (cid:0) Ω P | σ n − | +4 ( p ) e µ (cid:1) is one of Q -rings.Since Ω P | σ n − | +4 ( p ) is highly connected, π (cid:0) Ω P | σ n − | +4 ( p ) e µ (cid:1) ∼ = π ( R ). In particular, there isan E ∞ -map H π (cid:0) Ω P | σ n − | +4 ( p ) e µ (cid:1) → H F p . The compositeΩ P | σ n − | +4 ( p ) e µ → H π (cid:16) Ω P | σ n − | +4 ( p ) e µ (cid:17) → H F p is therefore a Q -algebra map. Moreover, the commosite R → Ω P | σ n − | +4 ( p ) e µ → H π (cid:16) Ω P | σ n − | +4 ( p ) e µ (cid:17) → H F p is simply the Postnikov truncation for R . It follows that the indecomposables in H ∗ (Ω P | σ n − | +4 ( p ) e µ )which come from the indecomposables in H ∗ ( R ) are sent to the indecomposables in H ∗ (H F p ).Using the discussion in the previous paragraph, Steinberger’s calculation (Theorem 2.2.4), andthe Dyer-Lashof hopping argument of Remark 2.2.9, we may conclude that the Steenrod comod-ule structure on H ∗ ( K µn ) (which, recall, is abstractly isomorphic to H ∗ (Θ( R ))) agrees with theSteenrod comodule structure on H ∗ (Θ( R )). Step 3.
By Step 2, the mod p homology of the Thom spectrum K µn is isomorphic to the mod p homology of the associated designer chromatic spectrum Θ( R ) as a Steenrod comodule. Themain result of [AL17] and [AP76, Theorem 1.1] now imply that unless R = B , the Thomspectrum K µn is equivalent to Θ( R ) upon p -completion for every prime p . Finally, if ConjectureF is true, then the same conclusion can be drawn for B : the Thom spectrum K µn is equivalentto Θ( B ) = tmf upon p -completion for every prime p .This concludes the proof of Theorem A. .3. Remark on the proof.
Before proceeding, we note the following consequence of the proofof Theorem A.
Proposition 5.3.1.
Let p be an odd prime. Assume Conjecture D and Conjecture E. Then thecomposite g : Ω S | σ n − | +3 → Ω P | σ n − | +4 ( p ) e µ −→ BGL ( X ( p n − → BGL (BP h n − i ) is null.Proof. Let R = X ( p n − R ) = BP h n − i . The map g is the composite withBGL ( Z ( R )) → B GL (Θ( R )) with the extension of the map σ n − : S | σ n − | +1 → BGL ( Z ( R ))along the double suspension S | σ n − | +1 → Ω S | σ n − | +3 . Since σ n − is null in π ∗ Θ( R ), we wouldbe done if g was homotopic to the dotted extension S | σ n − | +1 σ n − / / (cid:15) (cid:15) BGL (Θ( R ))Ω S | σ n − | +3 g ′ ♥♥♥♥♥♥ The potential failure of these maps to be homotopic stems from the fact that the composite Z ( R ) → R → Θ( R ) need not be a map of E -rings. It is, however, a map of E -rings; therefore,the maps g : Ω S | σ n − | +2 → BGL ( Z ( R )) → BGL (Θ( R ))and g ′ : Ω S | σ n − | +2 → BGL (Θ( R ))obtained by extending along the suspension S | σ n − | +1 → Ω S | σ n − | +2 are homotopic. We nowutilize the following result of Serre’s: Proposition 5.3.2 (Serre) . Let p be an odd prime. Then the suspension S n − → Ω S n splitsupon p -localization: there is a p -local equivalence E × Ω[ ι n , ι n ] : S n − × Ω S n − → Ω S n . This implies that the suspension map Ω S | σ n − | +2 → Ω S | σ n − | +3 admits a splitting as loopspaces. In particular, this implies that the map g is homotopic to the compositeΩ S | σ n − | +3 → Ω S | σ n − | +2 g −→ BGL ( Z ( R )) → BGL (Θ( R )) , and similarly for g ′ . Since g and g ′ are homotopic, and g ′ (and hence g ′ ) is null, we find that g is also null, as desired. (cid:3) Infinite families and the nilpotence theorem.
We now briefly discuss the relationshipbetween Theorem A and the nilpotence theorem. We begin by describing a special case ofthis connection. Recall from Remark 2.2.3 that Theorem 2.2.1 implies that if R is an E -ringspectrum, and x ∈ π ∗ R is a simple p -torsion element which has trivial MU-Hurewicz image,then x is nilpotent. A similar argument implies the following. Proposition 5.4.1.
Assume Conjecture D when n = 1 . Then Corollary 2.2.2 (i.e., Theorem Awhen n = 1 ) implies that if R is a p -local E -ring spectrum, and x ∈ π ∗ R is a class with trivial H Z p -Hurewicz image such that: • α x = 0 in π ∗ R ; and • the Toda bracket h p, α , x i contains zero;then x is nilpotent. roof. We claim that the composite(5.2) Ω S h i → BGL ( S ( p ) ) → BGL ( R [1 /x ])is null. Remark 4.1.8 implies that Conjecture D for n = 1 reduces us to showing that thecomposite Ω P p +1 ( p ) α −→ BGL ( S ( p ) ) → BGL ( R [1 /x ])is null. Since this composite is one of double loop spaces, it further suffices to show that thecomposite(5.3) P p − ( p ) → BGL ( S ( p ) ) → BGL ( R [1 /x ])is null. The bottom cell S p − of P p − ( p ) maps trivially to BGL ( R [1 /x ]), because the bottomcell detects α (by Remark 4.1.8), and α is nullhomotopic in R [1 /x ]. Therefore, the map (5.3)factors through the top cell S p − of P p − ( p ). The resulting map S p − → BGL ( S ( p ) ) → BGL ( R [1 /x ])detects an element of the Toda bracket h p, α , x i , but this contains zero by hypothesis, so isnullhomotopic.Since the map (5.2) is null, Corollary 2.2.2 and Theorem 2.1.7 implies that there is a ringmap H Z p → R [1 /x ]. In particular, the composite of the map x : Σ | x | R → R with the unit R → R [1 /x ] factors as shown:Σ | x | R x / / (cid:15) (cid:15) R / / (cid:15) (cid:15) R [1 /x ] . H Z p ∧ Σ | x | R x / / H Z p ∧ R ttttt The bottom map, however, is null, because x has zero H Z p -Hurewicz image. Therefore, theelement x ∈ π ∗ R [1 /x ] is null, and hence R [1 /x ] is contractible. (cid:3) Remark 5.4.2.
One can prove by a different argument that Proposition 5.4.1 is true withoutthe assumption that Conjecture D holds when n = 1. At p = 2, this was shown by Astey in[Ast97, Theorem 1.1].To discuss the relationship between Theorem A for general n and the nilpotence theorem(which we will expand upon in future work), we embark on a slight digression. The followingproposition describes the construction of some infinite families. Proposition 5.4.3.
Let R be a height n − spectrum as in the second line of Table 2, andassume Conjecture E if R = A or B . Then there is an infinite family σ n − ,p k ∈ π p k | v n |− ( R ) .Conjecture E implies that σ n − ,p k lifts to π p k | v n |− ( Z ( R )) , where Z ( R ) abusively denotes the E -center of X ( p n − if R = T ( n − .Proof. We construct this family by induction on k . The element σ n − , is just σ n − , so assumethat we have defined σ n − ,p k . The element σ n − ,p k ∈ π p k | v n |− R defines a map σ n − ,p k : S p k | v n | → BGL ( R ). When R = T ( n − σ n factors through the map BGL ( X ( p n − → BGL ( T ( n − R = A or B , Conjecture E (and the inductive hypothesis) implies that the map definedby σ n factors through the map BGL ( Z ( R )) → BGL ( R ). This implies that for all R as inthe second line of Table 2, the map σ n − ,p k : S p k | v n | → BGL ( R ) factors through an E -space,which we shall just denote by Z R for the purpose of this proof. If we assume Conjecture E, thenwe may take Z R = BGL ( Z ( R )). herefore, we get a map σ n − ,p k : Ω S p k | v n | +1 → BGL ( R ) via the compositeΩ S p k | v n | +1 → Z R → BGL ( R ) . Since Z R is an E -space, the map Ω S p k | v n | +1 → Z R is adjoint to a map _ j ≥ S jp k | v n | +1 ≃ ΣΩ S | v n | +1 → B Z R ;the source splits as indicated via the James splitting. These splittings are given by White-head products; in particular, the map S p k +1 | v n | +1 = S p ( p k | v n | +1) − ( p − → B Z R is given bythe p -fold Whitehead product [ σ n − ,p k , · · · , σ n − ,p k ]. This is divisible by p , so it yields a map S p k +1 | v n | → Z R , and hence a map S p k +1 | v n | → BGL ( R ) given by composing with the map Z R → BGL ( R ). This defines the desired element σ n − ,p k +1 ∈ π p k +1 | v n |− ( R ). As the construc-tion makes clear, assuming Conjecture E and taking Z R = BGL ( Z ( R )) implies that σ n − ,p k lifts to π p k | v n |− ( Z ( R )). (cid:3) Remark 5.4.4.
This infinite family is detected in the 1-line of the ANSS for R by δ ( v kn ), where δ is the boundary map induced by the map Σ − R/p → R . Remark 5.4.5.
The element σ n − , ∈ π p n − ( R ) is precisely σ n − . Remark 5.4.6.
When n = 1, the ring R is the ( p -local) sphere spectrum. The infinite family σ n − ,p k is the Adams-Toda α -family.We now briefly sketch an argument relating Theorem A to the proof of the nilpotence theorem;we shall elaborate on this discussion in forthcoming work. Remark 5.4.7.
The heart of the nilpotence theorem is what is called Step III in [DHS88];this step amounts to showing that certain self-maps of T ( n − G k in [DHS88]) of T ( n ) are nilpotent. These self-maps are given by multiplication by b n,k := σ n − ,p k ∈ π p k | v n |− T ( n −
1) at p = 2, and by multiplication by the p -fold Toda bracket b n,k = h σ n − ,p k , · · · , σ n − ,p k i at an odd prime p . (When p = 2, the element σ n − ,p k is denoted by h in[Hop87, Theorem 3].) It therefore suffices to establish the nilpotency of the b n,k .This can be proven through Theorem A via induction on k ; we shall assume ConjectureD and Conjecture E for the remainder of this discussion. The motivation for this approachstems from the observation that if R is any E - F p -algebra and x ∈ π ∗ ( R ), then there is arelation Q ( x ) = Q ( x ) (at odd primes, one has a relation involving the p -fold Toda bracket h Q ( x ) , · · · , Q ( x ) i ). In our setting, Proposition 5.4.3 implies that the elements σ n − ,k liftto π ∗ Z ( X ( p n − p = 2, one can prove (in the same way that the Cartan relation Q ( x ) = Q ( x ) is proven) that the construction of this infinite family implies that σ n − ,p k +1 can be described in terms of Q ( σ n − ,p k ). At odd primes, there is a similar relation involvingthe p -fold Toda bracket defining b n,k . In particular, induction on k implies that the b n,k are allnilpotent in π ∗ Z ( X ( p n − b n, is nilpotent.To argue that b n, is nilpotent, one first observes that σ n − b pn, = 0 in π ∗ Z ( X ( p n − n = 0, this follows from the statement that α β p = 0 in the sphere. To show that b n, is nilpotent, it suffices to establish that Z ( X ( p n − /b pn, ] is contractible; when n = 0, thisfollows from Proposition 5.4.1. We give a very brief sketch of this nilpotence for general n , byarguing as in Proposition 5.4.1, and with a generous lack of precision which will be remedied inforthcoming work.For notational convenience, we now denote d n, = b pn, . It suffices to show that the multiplication-by- d n, map d n, : Σ | d n, | Z ( X ( p n − → Z ( X ( p n − /d n, ] s nullhomotopic. Since σ n − kills d n, , we know that σ n − is nullhomotopic in Z ( X ( p n − /d n, ]. Moreover, the bracket h p, σ n − , Z ( X ( p n − /d n, ] i contains zero. By arguing as inProposition 5.4.1, we can conclude that the composite K n → Ω P | σ n − | +4 φ −→ BGL ( Z ( X ( p n − → BGL ( Z ( X ( p n − /d n ])is nullhomotopic, where the map φ is constructed in Step 1 of the proof of Theorem A. (Recallthat the proof of Theorem A shows that the Thom spectrum (Ω P | σ n − | +4 ) φ is an E ⊗ Q - Z ( X ( p n − h n − i splits off its base change along the map Z ( X ( p n − → T ( n − d n, map factors asΣ | d n, | Z ( X ( p n − d n, / / (cid:15) (cid:15) Z ( X ( p n − / / (cid:15) (cid:15) Z ( X ( p n − /d n, ] . Σ | d n, | (Ω P | σ n − | +4 ) φ d n, / / (Ω P | σ n − | +4 ) φ ❦❦❦❦❦❦❦ To show that the top composite is null, it therefore suffices to show that the self map of K φn defined by d n, is nullhomotopic. This essentially follows from the fact that (Ω P | σ n − | +4 ) φ isan E ⊗ Q - Z ( X ( p n − d n, is therefore null on K φn , because d n, is built from σ n − (which is null in (Ω P | σ n − | +4 ) φ ) via E -power operations.6. Applications
Splittings of cobordism spectra.
The goal of this section is to prove the following.
Theorem C.
Assume that the composite Z ( B ) → B → MString is an E -map. Then Con-jectures D, E, and F imply that there is a unital splitting of the Ando-Hopkins-Rezk orientation MString (2) → tmf (2) . Remark 6.1.1.
We believe that the assumption that the composite Z ( B ) → B → MString isan E -map is too strong: we believe that it can be removed using special properties of fibers ofcharming maps, and we will return to this in future work.We only construct unstructured splittings; it seems unlikely that they can be refined tostructured splittings. A slight modification of our arguments should work at any prime. Remark 6.1.2.
In fact, the same argument used to prove Theorem C shows that if the composite Z ( A ) → A → MSpin is an E -map, then Conjecture D and Conjecture E imply that thereare unital splittings of the Atiyah-Bott-Shapiro orientation MSpin → bo. This splitting wasoriginally proved unconditionally (i.e., without assuming Conjecture D or Conjecture E) byAnderson-Brown-Peterson in [ABP67] via a calculation with the Adams spectral sequence. Remark 6.1.3.
The inclusion of the cusp on M ell defines an E ∞ -map c : tmf → bo as in [LN14,Theorem 1.2]. The resulting diagramMString / / (cid:15) (cid:15) MSpin (cid:15) (cid:15) tmf c / / bocommutes (see, e.g., [Dev19b, Lemma 6.4]). The splitting s : tmf → MString of Theorem Cdefines a composite tmf s −→ MString → MSpin → bowhich agrees with c . emark 6.1.4. The Anderson-Brown-Peterson splitting implies that if X is any compactspace, then the Atiyah-Bott-Shapiro ˆ A -genus (i.e., the index of the Dirac operator in fami-lies) MSpin ∗ ( X ) → bo ∗ ( X ) is surjective. Similarly, if the composite Z ( B ) → B → MString isan E -map, then Conjectures D, E, and F imply that the Ando-Hopkins-Rezk orientation (i.e.,the Witten genus in families) MString ∗ ( X ) → tmf ∗ ( X ) is also surjective. Remark 6.1.5.
In [Dev19b], we proved (unconditionally) that the map π ∗ MString → π ∗ tmfis surjective. Our proof proceeds by showing that the map π ∗ B → π ∗ tmf is surjective viaarguments with the Adams-Novikov spectral sequence and by exploiting the E -ring structureon B to lift the powers of ∆ living in π ∗ tmf.The discussion preceding [MR99, Remark 7.3] implies that for a particular model of tmf (3),we have: Corollary 6.1.6.
Assume that the composite Z ( B ) → B → MString is an E -map. ThenConjectures D, E, and F imply that Σ tmf (3) is a summand of MString ∧ . We now turn to the proof of Theorem C.
Proof of Theorem C.
First, note that such a splitting exists after rationalization. Indeed, it suf-fices to check that this is true on rational homotopy; since the orientations under considerationsare E ∞ -ring maps, the induced map on homotopy is one of rings. It therefore suffices to lift thegenerators.We now show that the generators of π ∗ tmf ⊗ Q ∼ = Q [ c , c ] lift to π ∗ MString ⊗ Q . Although onecan argue this by explicitly constructing manifold representatives (as is done for c in [Dev19b,Corollary 6.3]), it is also possible to provide a more homotopy-theoretic proof: the elements c and c live in dimensions 8 and 12 respectively, and the map MString → tmf is known to be anequivalence in dimensions ≤
15. It follows that the same is true rationally, so c and c indeedlift to π ∗ MString ⊗ Q , as desired.We will now construct a splitting after p -completion (where p = 2). By Corollary 5.1.6, weobtain a unital map tmf ≃ Θ( B ) → MString upon p -completion which splits the orientationMString → Θ( B ) because:(a) the map Z ( B ) → B → MString is an E -ring map (by assumption).(b) the element σ vanishes in π MString (2) (because π MString (2) ∼ = π tmf (2) ∼ = 0),(c) and the Toda bracket h , σ , MString (2) i ⊆ π ∗ MString (2) contains zero (because π MString (2) ∼ = π tmf (2) , and the corresponding bracket contains 0 in π ∗ tmf (2) ).To obtain a map tmf ( p ) → MString ( p ) , we need to show that the induced map tmf ⊗ Q → tmf ∧ p ⊗ Q → MString ∧ p ⊗ Q agrees with the rational splitting constructed in the previousparagraph. However, this is immediate from the fact that the splittings tmf ∧ p → MString ∧ p areconstructed to be equivalences in dimensions ≤
15, and the fact that the map out of tmf ⊗ Q isdetermined by its effect on the generators c and c . (cid:3) Remark 6.1.7.
The proof recalled in the introduction of Thom’s splitting of MO proceededessentially unstably: there is an E -map Ω S → BO of spaces over BGL ( S ), whose Thomi-fication yields the desired E -map H F → MO. This argument also works for MSO: there isan E -map Ω S h i → BSO of spaces over BGL ( S ), whose Thomification yields the desired E -map H Z → MSO. One might hope for the existence of a similar unstable map which wouldyield Theorem C. We do not know how to construct such a map. To illustrate the difficulty,let us examine how such a proof would work; we will specialize to the case of MString, but thediscussion is the same for MSpin.According to Theorem A, Conjecture D and Conjecture E imply that there is a map K → BGL ( B ) whose Thom spectrum is equivalent to tmf. There is a map BN → B String, whose ber we will denote by Q . Then there is a fiber sequence N → BString → Q, and so Proposition 2.1.6 implies that there is a map Q → BGL ( B ) whose Thom spectrum isMString. Theorem C would follow if there was a map f : K → Q of spaces over BGL ( B ),since Thomification would produce a map tmf → MString.Conjecture D reduces the construction of f to the construction of a map Ω P (2) → Q . Thismap would in particular imply the existence of a map P (2) → Q (and would be equivalent tothe existence of such a map if Q was a double loop space), which in turn stems from a 2-torsionelement of π ( Q ). The long exact sequence on homotopy runs · · · → π (BString) → π ( Q ) → π ( N ) → π (BString) → · · · Bott periodicity states that π BString ∼ = π BString ∼ = 0, so we find that π ( Q ) ∼ = π ( N ).The desired 2-torsion element of π ( Q ) is precisely the element of π ( N ) described in Remark3.2.21. Choosing a particular nullhomotopy of twice this 2-torsion element of π ( Q ) producesa map g : P (2) → Q . To extend this map over the double suspension P (2) → Ω P (2), itwould suffice to show that there is a double loop space e Q with a map e Q → Q such that g factorsthrough e Q .Unfortunately, we do not know how to prove such a result; this is the unstable analogue ofConjecture E. In fact, such an unstable statement would bypass the need for Conjecture E inTheorem A. (One runs into the same obstruction for MSpin, except with the fiber of the map S → B Spin.) These statements are reminiscent of the conjecture (see Section 4.1) that thefiber W n = fib( S n − → Ω S n +1 ) of the double suspension admits the structure of a doubleloop space. Remark 6.1.8.
The following application of Theorem C was suggested by Mike Hopkins. In[HH92], the Anderson-Brown-Peterson splitting is used to show that the Atiyah-Bott-Shapiroorientation MSpin → KO induces an isomorphismMSpin ∗ ( X ) ⊗ MSpin ∗ KO ∗ ∼ = −→ KO ∗ ( X )of KO ∗ -modules for all spectra X . In future work, we shall show that Theorem C can be usedto prove the following height 2 analogue of this result: namely, Conjectures D, E, and F implythat the Ando-Hopkins-Rezk orientation MString → Tmf induces an isomorphism(6.1) MString ∗ ( X ) ⊗ MString ∗ Tmf ∗ ∼ = −→ Tmf ∗ ( X )of Tmf ∗ -modules for all spectra X . The K (1)-analogue of this isomorphism was obtained byLaures in [Lau04].6.2. Wood equivalences.
The Wood equivalence states that bo ∧ Cη ≃ bu. There are gen-eralizations of this equivalence to tmf (see [Mat16]); for instance, there is a 2-local 8-cell com-plex DA whose cohomology is isomorphic to the double of A (1) as an A (2)-module suchthat tmf (2) ∧ DA ≃ tmf (3) ≃ BP h i . Similarly, if X denotes the 3-local 3-cell complex S ∪ α e ∪ α e , then tmf (3) ∧ X ≃ tmf (2) ≃ BP h i ∨ Σ BP h i . We will use the umbrellaterm “Wood equivalence” to refer to equivalences of this kind.Our goal in this section is to revisit these Wood equivalences using the point of view stemmingfrom Theorem A. In particular, we will show that these equivalences are suggested by theexistence of certain EHP sequences; we find this to be a rather beautiful connection betweenstable and unstable homotopy theory. Each of the statements admit elementary proofs whichonly use the calculations in Section 3, but we will nonetheless spend a lot of time discussingtheir connections to EHP sequences. he first Wood-style result was proved in Proposition 3.2.5. The next result, originally provedin [Mah79, Section 2.5] and [DM81, Theorem 3.7], is the simplest example of a Wood-styleequivalence which is related to the existence of certain EHP sequences. Proposition 6.2.1.
Let S //η = X (2) (resp. S // ) denote the E -quotient of S by η (resp. ). If Y = Cη ∧ S / and A is a spectrum whose cohomology is isomorphic to A (1) as a module overthe Steenrod algebra, then there are equivalences A ∧ Cη ≃ S //η, A ∧ Y ≃ S // , A ∧ A ≃ y (1) /v of A -modules. Remark 6.2.2.
Proposition 6.2.1 implies the Wood equivalence bo ∧ Cη ≃ bu. Althoughthis implication is already true before 2-completion, we will work in the 2-complete categoryfor convenience. Recall that Theorem A states that Conjecture D and Conjecture E imply thatthere is a map µ : K → BGL ( A ) whose Thom spectrum is equivalent to bo (as left A -modules).Moreover, the Thom spectrum of the composite K µ −→ BGL ( A ) → BGL ( T (1)) is equivalentto BP h i . Since this Thom spectrum is the base-change K µ ∧ A T (1), and Proposition 6.2.1implies that T (1) = X (2) ≃ A ∧ Cη , we find thatBP h i ≃ K µ ∧ A ( A ∧ Cη ) ≃ K µ ∧ Cη ≃ bo ∧ Cη, as desired. Similarly, noting that S // y (1), we find that Proposition 6.2.1 also proves theequivalence bo ∧ Y ≃ k (1). Remark 6.2.3.
The argument of Remark 6.2.2 in fact proves that Theorem A for A impliesTheorem A for T (1), y Z (1), and y (1). Remark 6.2.4.
There are EHP sequences S → Ω S → Ω S , S → Ω S → Ω S . Recall that S / Cη , S // S //η = X (2), and A are Thom spectra over S , S , Ω S , Ω S , andΩ S respectively. Proposition 2.1.6 therefore implies that there are maps f : Ω S → B Aut( S / g : Ω S → B Aut( Cη ) whose Thom spectra are equivalent to S // S //η , respectively.The maps f and g define local systems of spectra over Ω S and Ω S whose fibers are equivalentto S / Cη (respectively), and one interpretation of Proposition 6.2.1 is that these localsystems in fact factor asΩ S η −→ BGL ( S ) → B Aut( S / , Ω S ν −→ BGL ( S ) → B Aut( Cη ) . If one could prove this directly, then Proposition 6.2.1 would be an immediate consequence. Weargue this for the first case in Remark 6.2.5. We do not know how to argue this for the secondcase.
Remark 6.2.5.
The first EHP sequence in Remark 6.2.4 splits via the Hopf map S → S . Themap f : Ω S → B Aut( S /
2) in fact factors through the dotted map in the following diagram: S / / Ω S (cid:15) (cid:15) / / Ω S (cid:15) (cid:15) w w ♣ ♣ ♣ ♣ ♣ ♣ BGL ( S ) / / B Aut( S / . Indeed, the composite Ω S → Ω S → BGL ( S ) is a loop map, and therefore is determined bythe composite φ : S → S → B GL ( S ). Since the map S → B GL ( S ) detects the element − ∈ π ( S ) × , the map φ does in fact determine a unit multiple of η . This implies the desiredclaim. roof of Proposition 6.2.1. For the first two equivalences, it suffices to show that A ∧ Cη ≃ S //η and that S //η ∧ S / ≃ S //
2. We will prove the first statement; the proof of the second statementis exactly the same. There is a map Cη → S //η given by the inclusion of the 2-skeleton. Thereis also an E -ring map A → S //η given as follows. The multiplication on S //η defines a unitalmap Cη ∧ Cη → S //η . But since the Toda bracket h η, , η i contains ν , there is a unital map Cν → Cη ∧ Cη . This supplies a unital map Cν → S //η , which, by the universal property of A = S //ν (via Theorem 2.1.7), extends to an E -ring map A → S //η .For the final equivalence, it suffices to construct a map A → y (1) /v for which the inducedmap A ∧ A → y (1) /v gives an isomorphism on mod 2 homology. Since A may be obtainedas the cofiber of a v -self map Σ Y → Y , it suffices to observe that the the following diagramcommutes; our desired map is the induced map on vertical cofibers:Σ Y v (cid:15) (cid:15) / / Σ y (1) v (cid:15) (cid:15) Y / / y (1) . (cid:3) Next, we have the following result at height 2:
Proposition 6.2.6.
Let DA denote the double of A (see [Mat16] ). There are -completeequivalences B ∧ DA ≃ T (2) , B ∧ Z ≃ y (2) , B ∧ A ≃ y (2) /v , where Z is the spectrum “ A ” from [MT94, BE16] , and A is a spectrum whose cohomologyis isomorphic to A (2) as a module over the Steenrod algebra. Remark 6.2.7.
Arguing as in Remark 6.2.2 shows that Proposition 6.2.6 and Theorem A implythe Wood equivalencestmf ∧ DA ≃ tmf (3) = BP h i , tmf ∧ Z ≃ k (2) , tmf ∧ A ≃ H F . Remark 6.2.8.
Exactly as in Remark 6.2.3, the argument of Remark 6.2.7 in fact proves thatTheorem A for B implies Theorem A for T (2), y Z (2), and y (2). Remark 6.2.9.
The telescope conjecture, which we interpet as stating that L n -localization isthe same as L fn -localization, is known to be true at height 1. For odd primes, it was proved byMiller in [Mil81], and at p = 2 it was proved by Mahowald in [Mah81a, Mah82]. Mahowald’sapproach was to calculate the telescopic homotopy of the type 1 spectrum Y . In [MRS01],Mahowald-Ravenel-Shick proposed an approach to dis proving the telescope conjecture at height2: they suggest that for n ≥
2, the L n -localization and the v n -telescopic localization of y ( n ) havedifferent homotopy groups. They show, however, that the L -localization and the v -telescopiclocalization of y (1) agree, so this approach (thankfully) does not give a counterexample to thetelescope conjecture at height 1.Motivated by Mahowald’s approach to the telescope conjecture, Behrens-Beaudry-Bhattacharya-Culver-Xu study the v -telescopic homotopy of Z in [BBB + R denote A or B . Moreover, let F denote Y or Z (dependingon what R is), and let R ′ denote y (1) or y (2) (again depending on what R is), so that R ∧ F = R ′ by Propositions 6.2.1 and 6.2.6. Then: In the former source, Z is denoted by M . orollary 6.2.10. If the telescope conjecture is true for F (and hence any type or spectrum)or R , then it is true for R ′ .Proof. Since L n - and L fn -localizations are smashing, we find that if the telescope conjecture istrue for F or R , then Propositions 6.2.1 and 6.2.6 yield equivalences L fn R ′ ≃ R ∧ L fn F ≃ R ∧ L n F ≃ L n R ′ . (cid:3) Finally, we prove Proposition 6.2.6.
Proof of Proposition 6.2.6.
We first construct maps B → T (2) and DA → T (2). The topcell of DA is in dimension 12, and the map T (2) → BP is an equivalence in dimensions ≤
12. It follows that constructing a map DA → T (2) is equivalent to constructing a map DA → BP. However, both BP and DA are concentrated in even degrees, so the Atiyah-Hirzebruch spectral sequence collapses, and we find that BP ∗ ( DA ) ∼ = H ∗ ( DA ; BP ∗ ). Thegenerator in bidegree (0 ,
0) produces a map DA → T (2); its effect on homology is the additiveinclusion F [ ζ , ζ ] / ( ζ , ζ ) → F [ ζ , ζ ].The map B → T (2) may be defined via the universal property of Thom spectra from Section2.1 and Remark 3.2.17. Its effect on homology is the inclusion F [ ζ , ζ ] → F [ ζ , ζ ]. Weobtain a map B ∧ DA → T (2) via the multiplication on T (2), and this induces an isomorphismin mod 2 homology.For the second equivalence, we argue similarly: the map B → T (2) defines a map B → T (2) → y (2). Next, recall that Z is built through iterated cofiber sequences:Σ Y v −→ Y → A , Σ A ∧ Cν σ −→ A ∧ Cν → Z. As an aside, we note that the element σ is intimately related to the element discussed inExample 3.1.14; namely, it is given by the self-map of A ∧ Cν given by smashing A with thefollowing diagram: Σ Cν σ + + ❲❲❲❲❲❲❲❲❲❲❲❲❲❲ / / Σ A σ ∧ id / / A ∧ A / / ACν. O O Using these cofiber sequences and Proposition 3.2.5, one obtains a map Z → y (2), which inducesthe additive inclusion F [ ζ , ζ ] / ( ζ , ζ ) → F [ ζ , ζ ] on mod 2 homology. The multiplicationon y (2) defines a map B ∧ Z → y (2), which induces an isomorphism on mod 2 homology.For the final equivalence, it suffices to construct a map A → y (2) /v for which the inducedmap B ∧ A → y (2) /v gives an isomorphism on mod 2 homology. Since A may be obtainedas the cofiber of a v -self map Σ Z → Z , it suffices to observe that the the following diagramcommutes; our desired map is the induced map on vertical cofibers:Σ Z v (cid:15) (cid:15) / / Σ y (2) v (cid:15) (cid:15) Z / / y (2) . (cid:3) Arguing exactly as in the proof of Proposition 6.2.6 shows the following result at the prime3: roposition 6.2.11. Let X denote the -skeleton of T (1) = S //α . There are -completeequivalences B ∧ X ≃ T (2) ∨ Σ T (2) , B ∧ X ∧ S / (3 , v ) ≃ y (2) ∨ Σ y (2) . We now discuss the relation between Proposition 6.2.6 and EHP sequences, along the linesof Remark 6.2.4. We begin by studying the spectrum y Z (2), which is a spectrum such thatmodding out by 2 yields y (2). The spectrum y Z (2) is the Thom spectrum of a map Ω J ( S ) h i ,but we will need a more explicit description of the space Ω J ( S ) h i . Remark 6.2.12.
By [Ame18, Proposition 4.3], there is a fiber sequence(6.2) V ( R ) ≃ J ( S ) h i → J ( S ) → C P ∞ , where V k ( R n ) is the Stiefel manifold of orthonormal k -frames in R n . The fiber sequence (6.2)is split upon looping, so we obtain an equivalence Ω J ( S ) ≃ Ω V ( R ) × S . Recall that y (2)arises as the Thom spectrum of a map Ω J ( S ) → BGL ( S ) such that the composite with S → Ω J ( S ) detects − ∈ π ( S ) × . It follows from Proposition 2.1.6 that there is a map φ : Ω V ( R ) → B Aut( S /
2) whose Thom spectrum is y (2). By arguing as in Remark 6.2.5, wefind that φ factors as Ω V ( R ) µ −→ BGL ( S ) → B Aut( S / µ : Ω V ( R ) → BGL ( S ). It follows that y (2) ≃ Ω V ( R ) µ ∧ S / ≃ y Z (2) ∧ S / . Construction 6.2.13.
The map µ admits the following explicit description. Recall that V ( R ) sits in a fiber sequence S → V ( R ) → S , so taking loops defines a fiber sequence with total space Ω V ( R ). We now define a real vectorbundle over Ω V ( R ) via the map of fiber sequencesΩ V ( R ) / / λ (cid:15) (cid:15) Ω S / / (cid:15) (cid:15) S (cid:15) (cid:15) BO / / ∗ / / B O , where the map S → B O detects the generator of π (BO) ∼ = Z /
2. The map µ is then thecomposite of λ with the J-homomorphism BO → BGL ( S ).Next, we turn to the relationship between T (2) and y Z (2). Construction 6.2.14.
Remark 3.1.9 says that T (2) is the Thom spectrum of a bundle overΩ V ( H ), and Remark 6.2.12 says that y Z (2) is the Thom spectrum of a bundle over Ω V ( R ).Just as in Remark 6.2.4, the equivalence T (2) /v ≃ y Z (2) of Proposition 3.2.5 is related to an“EHP” fiber sequence relating Ω V ( R ) and Ω V ( H ). To describe this fiber sequence, recallthat V ( H ) ≃ Sp(2), which is isomorphic to Spin(5). There is therefore a map V ( H ) ≃ Sp(2) ≃ Spin(5) → SO(5) → SO(5) / SO(3) ≃ V ( R ) . There is a pullback square V ( H ) / / (cid:15) (cid:15) S ν (cid:15) (cid:15) V ( R ) / / S , here both the horizontal and vertical fibers are S . This results in a fiber sequence(6.3) Ω V ( H ) → Ω V ( R ) → S ;this fiber sequence is in fact split, because loops on the Hopf map ν : S → S splits the Hopfmap Ω S → Ω S . The compositeΩ V ( H ) → Ω V ( R ) λ −→ BOagrees with the map µ : Ω V ( H ) → BO from Remark 3.1.9, and the Thom spectrum of thiscomposite is the canonical map T (2) → y Z (2). The map S → BGL (Ω V ( H ) µ ) = BGL ( T (2))arising via Proposition 2.1.6 detects v ∈ π T (2).We now discuss the relationship between T (2) and B . Remark 6.2.15.
Recall that Remark 3.1.9 says that T (2) is the Thom spectrum of a bundleover Ω V ( H ), and Definition 3.2.16 says that B is the Thom spectrum of a bundle over N . Wemay expect that there is a map of fiber sequences(6.4) Ω S / / H (cid:15) (cid:15) Ω V ( H ) / / (cid:15) (cid:15) Ω S H (cid:15) (cid:15) Ω S / / N / / Ω S . The fiber of the leftmost map is J ( S ), and the fiber of the rightmost map is S , so the fiberof the map Ω V ( H ) → N would sit in a fiber sequence J ( S ) → fib(Ω V ( H ) → N ) → S . We do not know how to construct a map of fiber sequences as in (6.4), but one possible approachis as follows. In [Hop84, Section 3.2], Hopkins proves that the Thom space ( C P n − ) L is a gen-erating complex for ΩSp( n ), where L is the tautological line bundle over C P n − . In particular,there is a map ( C P ) L → Ω V ( H ) whose image on homology generates H ∗ (Ω V ( H )). (Thehomology of ( C P ) L is free of rank one in degrees 2, 4, and 6.) One might therefore hope toconstruct a map Ω V ( H ) → N via a map ΩΣ( C P ) L → N which factors through Ω V ( H ).The map ΩΣ( C P ) L → N would be adjoint to a map _ k ≥ Σ(( C P ) L ) ∧ k ≃ ΣΩΣ( C P ) L → B N, where we have indicated the James splitting of the source. By connectivity, any map Σ( C P ) L → B N is null, but there may be a map Σ( C P ) L ∧ ( C P ) L → B N producing the desired mapΣΩΣ( C P ) L → B N . We leave this as an unresolved question.6.3. Topological Hochschild homology.
In this section, we calculate an approximation ofthe topological Hochschild homology of BP h n i , bo, and tmf. Definition 6.3.1.
The unstructured relative topological Hochschild homology of BP h n i , bo,and tmf over T ( n ), A , and B (respectively) is given by: [ THH T ( n ) (BP h n i ) ≃ BP h n i ∧ Σ ∞ + B K n +1 , [ THH A (bo) ≃ bo ∧ Σ ∞ + B K , [ THH B (tmf) ≃ tmf ∧ Σ ∞ + B K , where, as usual, K n +1 denotes the fiber of a charming map Ω S p n +1 +1 → S p n +1 − and K (resp. K ) denotes the fiber of a charming map Ω S → S (resp. Ω S → S ). Remark 6.3.2.
Theorem A implies that there are unital maps THH(BP h n i ) → [ THH T ( n ) (BP h n i ),THH(bo) → [ THH A (bo), and THH(tmf) → [ THH B (tmf). e then have the following calculation: Theorem 6.3.3.
The homotopy of [ THH T ( n ) (BP h n i ) is given by π ∗ [ THH T ( n ) (BP h n i ) ∼ = π ∗ BP h n i⊕ T ∗ , where T ∗ is a torsion graded abelian group concentrated in odd degrees, with a degreewisefiltration whose associated graded in degree d − is gr( T d − ) ∼ = M k ≥ π d − p n + k +1 − p n +1 + p k ) BP h n i /p k . Remark 6.3.4.
It is probably also possible to compute π ∗ [ THH A (bo) and π ∗ [ THH B (tmf), but,unlike Theorem 6.3.3, there are lots of possibilities for differentials in the Atiyah-Hirzebruchspectral sequence. Since we do not have a good description at the moment, we leave this asan unresolved question that we hope to return to in future work. It would be very interest-ing to determine the Hurewicz image of π ∗ THH(bo) in π ∗ [ THH A (bo) via the calculations in[AHL10, Section 7], and use it to make conjectures about the Hurewicz image of π ∗ THH(tmf)in π ∗ [ THH B (tmf). Remark 6.3.5.
The calculation of the additive extensions in the torsion subgroup of π ∗ [ THH T ( n ) (BP h n i )is quite complicated, so we have not attempted to give a complete description in the statementof Theorem 6.3.3. There may be a simpler combinatorial way to express the extensions. Remark 6.3.6.
The homotopy of THH(BP h i ) was calculated in [AHL10, Theorem 2.6], whereit was shown that π ∗ THH(BP h i ) ∼ = BP h i ∗ ⊕ BP h i ∗ " v p k + ··· + p p k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ≥ ⊕ M k ≥ p − M m =1 Σ mp k +2 +2( p − T k,m , where T k,m is an explicit torsion π ∗ BP h i -module with generators living in degrees 2 P ki =1 m i p n +2 − i where 0 ≤ m i ≤ p −
1, and with numerous relations. The copy of BP h i ∗ appearing as a sum-mand in π ∗ THH(BP h i ) is sent to the corresponding summand in π ∗ [ THH T (1) (BP h i ). One canalso attempt to compare the torsion summands. Theorem 6.3.3 gives a description of the torsionsummand of π ∗ [ THH T (1) (BP h i ) up to additive extension problems. Although we have not com-puted these additive extensions even in this case, we believe that the torsion in π ∗ THH(BP h i )injects into the torsion in π ∗ [ THH T (1) (BP h i ), and that this in fact true more generally at allheights (i.e., the torsion in π ∗ THH(BP h n i ) injects into the torsion in π ∗ [ THH T ( n ) (BP h n i )). Proof of Theorem 6.3.3.
Definition 6.3.1 gives an Atiyah-Hirzebruch spectral sequence E s,t = H s (B K n +1 ; π t BP h n i ) ⇒ π s + t [ THH T ( n ) (BP h n i ) , where the differentials go d r : E rs,t → E rs − r,t + r − . To calculate this spectral sequence, we useRemark 4.2.3 (after p -completion): this tells us that E s, ∗ ∼ = π ∗ BP h n i s = 0 π ∗ BP h n i /p k if s = 2( p k + 1) p n +1 − p k −
10 else . A generic element of this E -page lives in bidegree ( s, t ) = (2( p k + 1) p n +1 − p k − , m ) for m = P ni =1 m i ( p i −
1) with m i ∈ Z ≥ .We now need to compute the differentials in the spectral sequence. Notice that no differentialshit the s = 0-line. There are a few ways to see this: one way is to note that BP h n i splits off THH T ( n ) (BP h n i ); another way is to note that any differential that hits the 0-line originatesfrom a torsion group and lands in π ∗ BP h n i , and hence is automatically zero since the target istorsion-free.We claim that there are no differentials between the torsion groups; this is essentially fordegree reasons. Indeed, the d r -differential changes bidegree by ( − r, r − s, t ) = (2( p k + 1) p n +1 − p k − , d ) for k ≥
1, we find that anypossibly nonzero differential d r must have r = 2( p k − p ℓ )( p n +1 −
1) for some k, ℓ ≥
1. Thisdifferential would go from the (2( p k + 1) p n +1 − p k − p ℓ + 1) p n +1 − p ℓ − r must be even, the d r -differential must increase the t -degree by an odd integer.However, it is clear from the above description of the E -page that all elements have even t -degree, and hence any differential necessarily vanishes. In particular, the Atiyah-Hirzebruchspectral sequence collapses at the E -page. (cid:3) C -equivariant analogue of Corollary B Our goal in this section is to study a C -equivariant analogue of Corollary B at height 1. Theodd primary analogue of this result is deferred to the future; it is considerably more subtle.7.1. C -equivariant analogues of Ravenel’s spectra. In this section, we construct the C -equivariant analogue of T ( n ) for all n . We 2-localize everywhere until mentioned otherwise.There is a C -action on ΩSU( n ) given by complex conjugation, and the resulting C -space isdenoted ΩSU( n ) R . Real Bott periodicity gives a C -equivariant map ΩSU( n ) R → BU R whoseThom spectrum is the (genuine) C -spectrum X ( n ) R . This admits the structure of an E ρ -ring.As in the nonequivariant case, the equivariant Quillen idempotent on MU R restricts to one on X ( m ) R , and therefore defines a summand T ( n ) R of X ( m ) R for 2 n ≤ m ≤ n +1 −
1. Again, thissummand admits the structure of an E -ring. Construction 7.1.1.
There is an equivariant fiber sequenceΩSU( n ) R → ΩSU( n + 1) R → Ω S nρ +1 , where ρ is the regular representation of C ; the equivariant analogue of Proposition 2.1.6 thenshows that there is a map Ω S nρ +1 → BGL ( X ( n ) R ) (detecting an element χ n ∈ π nρ − X ( n ) R )whose Thom spectrum is X ( n + 1) R .If e σ n denotes the image of the element χ n +1 ρ − in π (2 n +1 − ρ − T ( n ) R , then we have a C -equivariant analogue of Lemma 3.1.12: Lemma 7.1.2.
The Thom spectrum of the map Ω S (2 n +1 − ρ +1 → BGL ( X (2 n +1 − R ) → B GL ( T ( n ) R ) detecting e σ n is T ( n + 1) R . Example 7.1.3.
For instance, T (1) R = X (2) R is the Thom spectrum of the map Ω S ρ +1 → BU R ; upon composing with the equivariant J-homomorphism BU R → BGL ( S ), this detects theelement e η ∈ π σ S , and the extension of the map S ρ → BGL ( S ) to Ω S ρ +1 uses the E -structureon BGL ( S ). The case of X (2) R exhibits a curious property: S ρ +1 is the loop space Ω σ H P ∞ R ,and there are equivalences (see [HW19])Ω S ρ +1 ≃ Ω σ +1 H P ∞ R ≃ Ω σ (Ω H P ∞ R ) . However, Ω H P ∞ R ≃ S ρ + σ , so Ω S ρ +1 = Ω σ S ρ + σ . The map Ω σ S ρ + σ → BGL ( S ) still detects theelement e η ∈ π σ S on the bottom cell, but the extension of the map S ρ → BGL ( S ) to Ω σ S ρ + σ is now defined via the E σ -structure on BGL ( S ). The upshot of this discussion is that X (2) R is not only the free E -ring with a nullhomotopy of e η , but also the free E σ -algebra with anullhomotopy of e η . arning 7.1.4. Unlike the nonequivariant setting, the element e η ∈ π σ S is neither torsion nornilpotent. This is because its geometric fixed points is Φ C e η = 2 ∈ π S . Example 7.1.5.
Consider the element e σ ∈ π ρ − T (1) R . The underlying nonequivariant ele-ment of π T (1) R is simply σ . To determine Φ C e σ ∈ π Φ C T (1) R , we first note that Φ C T (1) R is the Thom spectrum of the map Φ C e η : Φ C Ω S ρ +1 → BGL ( S ). Since Φ C Ω S ρ +1 = Ω S andΦ C e η = 2, we find that Φ C T (1) R is the E -quotient S //
2. The element Φ C e σ ∈ π S // ∼ = π S / S → S / η on the top cell. Such a map exists because 2 η = 0.As an aside, we mention that there is a C -equivariant lift of the spectrum A : Definition 7.1.6.
Let A C denote the Thom spectrum of the Real bundle over Ω S ρ +1 definedby the extension of the map S ρ → BSU R detecting a generator of π ρ BSU R . Remark 7.1.7.
The underlying spectrum of A C is A . To determine the geometric fixed pointsof A C , note that the equivariant J-homomorphism BU R → BGL ( S ) sends the generator of π ρ BU R to the equivariant Hopf map e ν ∈ π ρ − S . Therefore, Φ C A C is the Thom spectrum ofthe map Φ C e ν : Φ C Ω S ρ +1 → BGL ( S ). Since e Φ C e ν = η and Φ C Ω S ρ +1 = Ω S , we find thatΦ C A C = T (1). In particular, A C may be thought of as the free C -equivariant E -ring witha nullhomotopy of e ν . Example 7.1.8.
The element e σ lifts to π ρ − A C . Indeed, Remark 3.2.15 works equivariantlytoo: the equivariant Hopf map S ρ − → S ρ defines a composite S ρ − → S ρ → Ω S ρ +1 .The composite S ρ − → Ω S ρ +1 → BSU R is null, since π ρ − BSU R = 0. It follows thatupon Thomification, the map S ρ − → Ω S ρ +1 defines an element e σ ′ of π ρ − A C . In orderto show that this element indeed deserves to be called e σ , we use Proposition 7.1.9. The map A C → T (1) R from the proposition induces a map π ρ − A C → π ρ − T (1) R , and we need toshow that the image of e σ ′ ∈ π ρ − A C is in fact e σ . By Example 7.1.5, it suffices to observethat the underlying nonequivariant map corresponding to e σ ′ ∈ π ρ − T (1) R is σ , and that thegeometric fixed points Φ C e σ ′ ∈ π S // η appearing in Example 7.1.5.We now prove the proposition used above. Proposition 7.1.9.
There is a genuine C -equivariant E -map A C → T (1) R .Proof. By Example 7.1.7, it suffices to show that e ν = 0 ∈ π ρ − T (1) R . The underlying map isnull, because ν = 0 ∈ π T (1). The geometric fixed points are also null, because Φ C e ν = η isnull in π Φ C T (1) R = π S //
2. Therefore, e ν is null in π ρ − T (1) R . (cid:3) In fact, it is easy to prove the following analogue of Proposition 6.2.1:
Proposition 7.1.10.
There is a C -equivariant equivalence A C ∧ C e η ≃ T (1) R .Proof. There are maps A C → T (1) R and C e η → T (1) R , which define a map A C ∧ C e η → T (1) R via the multiplication on T (1) R . This map is an equivalence on underlying by Proposition 6.2.1,and on geometric fixed points induces the map T (1) ∧ S / → S //
2. This was also proved in thecourse of Proposition 6.2.1. (cid:3)
Remark 7.1.11.
As in Remark 6.2.2, one might hope that this implies the C -equivariantWood equivalence bo C ∧ C e η ≃ bu R via some equivariant analogue of Theorem A. Remark 7.1.12.
The equivariant analogue of Remark 6.2.4 remains true: the equivariant Woodequivalence of Proposition 7.1.10 stems from the EHP sequence S ρ → Ω S ρ +1 → Ω S ρ +1 . .2. The C -equivariant analogue of Corollary B at n = 1 . Recall (see [HK01]) thatthere are indecomposable classes v n ∈ π (2 n − ρ BP R ; as in Theorem 3.1.5, these lift to classes in π ⋆ T ( m ) R if m ≥ n . The main result of this section is the following: Theorem 7.2.1.
There is a map Ω ρ S ρ +1 → BGL ( T (1) R ) detecting an indecomposable in π ρ T (1) R on the bottom cell, whose Thom spectrum is H Z . Note that, as with Corollary B at n = 1, this result is unconditional . The argument is exactlyas in the proof of Corollary B at n = 1, with practically no modifications. We need the followinganalogue of Theorem 2.2.1, originally proved in [BW18, HW19]. Proposition 7.2.2 (Behrens-Wilson, Hahn-Wilson) . Let p be any prime, and let λ denote the -dimensional standard representation of C p on C . The Thom spectrum of the map Ω λ S λ +1 → BGL ( S ) extending the map − p : S → BGL ( S ) is equivalent to H F p as an E λ -ring.Moreover, if S λ +1 h λ + 1 i denotes the ( λ + 1) -connected cover of S λ +1 , then the Thom spectrumof the induced map Ω λ S λ +1 h λ + 1 i → BGL ( S ) is equivalent to H Z as an E λ -ring. We can now prove Theorem 7.2.1.
Proof of Theorem 7.2.1.
In [HW19], the authors prove that there is an equivalence of C -spacesbetween Ω λ S λ +1 and Ω ρ S ρ +1 , and that H F is in fact the Thom spectrum of the induced mapΩ ρ S ρ +1 → BGL ( S ) detecting −
1. Since both Ω ρ S ρ +1 h ρ + 1 i and Ω λ S λ +1 h λ + 1 i are definedas fibers of maps to S which are degree one on the bottom cell, Hahn and Wilson’s equivalencelifts to a C -equivariant equivalence Ω ρ S ρ +1 h ρ + 1 i ≃ Ω λ S λ +1 h λ + 1 i , and we find that H Z isthe Thom spectrum of the map Ω ρ S ρ +1 h ρ + 1 i → BGL ( S ).Since T (1) R is the Thom spectrum of the composite map Ω S ρ +1 → Ω ρ S ρ +1 h ρ + 1 i → BGL ( S ) detecting e η on the bottom cell of the source, it follows from the C -equivariantanalogue of Proposition 2.1.6 and the above discussion that it is sufficient to define a fibersequence Ω S ρ +1 → Ω ρ S ρ +1 h ρ + 1 i → Ω ρ S ρ +1 , and check that the induced map Ω ρ S ρ +1 → BGL ( T (1) R ) detects an indecomposable elementof π ρ T (1) R . See Remark 4.1.4 for the nonequivariant analogue of this fiber sequence.Since there is an equivalence Ω S ρ +1 ≃ Ω σ S ρ + σ , it suffices to prove that there is a fibersequence(7.1) S ρ + σ → Ω S ρ +1 h ρ + 1 i → Ω S ρ +1 ;taking σ -loops produces the desired fiber sequence. The fiber sequence (7.1) can be obtained bytaking vertical fibers in the following map of fiber sequences S ρ / / (cid:15) (cid:15) Ω S ρ +1 / / (cid:15) (cid:15) Ω S ρ +1 (cid:15) (cid:15) C P ∞ R C P ∞ R / / ∗ . Here, the top horizontal fiber sequence is the EHP fiber sequence S ρ → Ω S ρ +1 → Ω S ρ +1 . To identify the fibers, note that there is the Hopf fiber sequence S ρ + σ e η −→ S ρ → C P ∞ R . The fiber of the middle vertical map is Ω S ρ +1 h ρ + 1 i via the definition of S ρ +1 h ρ + 1 i as thehomotopy fiber of the map S ρ +1 → B C P ∞ R . t remains to show that the map Ω ρ S ρ +1 → BGL ( T (1) R ) detects an indecomposable ele-ment of π ρ T (1) R . Indecomposability in π ρ T (1) R ∼ = π ρ BP R is the same as not being divisibleby 2, so we just need to show that the dotted map in the following diagram does not exist: S ρ +1 E (cid:15) (cid:15) & & ◆◆◆◆◆◆◆◆◆◆◆ Ω ρ S ρ +1 / / (cid:15) (cid:15) S ρ +1 x x q q q q q q BGL ( T (1) R )If this factorization existed, there would be an orientation H Z → T (1) R , which is absurd. (cid:3) We now explain why we do not know how to prove the equivariant analogue of Corollary B athigher heights. One could propose an following equivariant analogue of Conjecture D, and sucha conjecture would obviously be closely tied with the existence of some equivariant analogue ofthe work of Cohen-Moore-Neisendorfer. We do not know if any such result exists, but it wouldcertainly be extremely interesting.Suppose that one wanted to prove a result like Corollary B, stating that the equivariant ana-logues of Conjecture D and Conjecture E imply that there is a map Ω ρ S n ρ +1 → BGL ( T ( n ) R )detecting an indecomposable in π (2 n − ρ T ( n ) R on the bottom cell, whose Thom spectrum isBP h n − i R . One could then try to run the same proof as in the nonequivariant case by con-structing a map from the fiber of a charming map Ω ρ S n ρ +1 → S (2 n − ρ +1 to BGL ( T ( n − R ),but the issue comes in replicating Step 1 of Section 5.2: there is no analogue of Lemma 3.1.16,since the equivariant element e σ n ∈ π ⋆ T ( n ) is neither torsion nor nilpotent. See Warning 7.1.4.This is intimately tied with the failure of an analogue of the nilpotence theorem in the equivari-ant setting. In future work, we shall describe a related project connecting the T ( n ) spectra tothe Andrews-Gheorghe-Miller w n -periodicity in C -motivic homotopy theory.However, since there is a map Ω λ S λ +1 h λ + 1 i → BGL ( S ) as in Proposition 7.2.2, theremay nevertheless be a way to construct a suitable map from the fiber of a charming mapΩ ρ S n ρ +1 → S (2 n − ρ +1 to BGL ( T ( n − R ). Such a construction would presumably providea more elegant construction of the nonequivariant map used in the proof of Theorem A.8. Future directions
In this section, we suggest some directions for future investigation. This is certainly not anexhaustive list; there are numerous questions we do not know how to address that are spatteredall over this document, but we have tried to condense some of them into the list below. We havetried to order the questions in order of our interest in them. We have partial progress on manyof these questions.(a) Some obvious avenues for future work are the conjectures studied in this paper: Con-jectures D, E, F, and 3.1.7. Can the E -assumption in the statement of Theorem C beremoved? Prove the existence of a map of fiber sequences as in Remark 6.2.15.(b) One of the Main Goals TM of this project is to rephrase the proof of the nilpotencetheorem from [DHS88, HS98]. As mentioned in Remark 2.2.3, the Hopkins-Mahowaldtheorem for H F p immediately implies the nilpotence theorem for simple p -torsion classesin the homotopy of a homotopy commutative ring spectrum (see also [Hop84]). We willexpand on the relation between the results of this paper and the nilpotence theorem inforthcoming work; see Remark 5.4.7 for a sketch. rom this point of view, Theorem A is very interesting: it connects torsion in theunstable homotopy groups of spheres (via Cohen-Moore-Neisendorfer) to nilpotence inthe stable homotopy groups of spheres. We are not sure how to do so, but could theCohen-Moore-Neisendorfer bound for the exponents of unstable homotopy groups ofspheres be used to obtain bounds for the nilpotence exponent of the stable homotopygroups of spheres?(c) It is extremely interesting to contemplate the interaction between unstable homotopytheory and chromatic homotopy theory apparent in this paper. Connections betweenunstable homotopy theory and the chromatic picture have appeared elsewhere in theliterature (e.g., in [AM99, Aro98, Mah82, MT94]), but their relationship to the contentof this project is not clear to me. It would be interesting to have this clarified. Onena¨ıve hope is that such connection could stem from a construction of a charming map(such as the Cohen-Moore-Neisendorfer map) via Weiss calculus.(d) Let R denote S or A . The map R → Θ( R ) is an equivalence in dimensions < | σ n | .Moreover, the Θ( R )-based Adams-Novikov spectral sequence has a vanishing line ofslope 1 / | σ n | (see [Mah81a] for the case R = A ). Can another proof of this vanishing linebe given using general arguments involving Thom spectra? We have some results in thisdirection which we shall address in future work.(e) The unit maps from each of the Thom spectra on the second line of Table 1 to thecorresponding designer spectrum on the third line are surjective on homotopy. In thecase of tmf, this requires some computational effort to prove, and has been completedin [Dev19b]. This behavior is rather unexpected: in general, the unit map from astructured ring to some structured quotient will not be surjective on homotopy. Is therea conceptual reason for this surjectivity?(f) In [BBB + Z is studied. Mahowalduses the Thom spectrum A to study the bo-resolution of the sphere in [Mah81a], soperhaps the spectrum B could be used to study the tmf-resolution of Z . This is workin progress. See also Corollary 6.2.10 and the discussion preceding it.(g) Is there an equivariant analogue of Theorem A at higher heights and other primes?Currently, we have such an analogue at height 1 and at p = 2; see Section 7.(h) The Hopkins-Mahowald theorem may used to define Brown-Gitler spectra. Theorem Aproduces “relative” Brown-Gitler spectra for BP h n i , bo, and tmf. In future work, wewill study these spectra and show how they relate to the Davis-Mahowald non -splittingof tmf ∧ tmf as a wedge of shifts of bo-Brown-Gitler spectra smashed with tmf from[DM10].(i) The story outlined in the introduction above could fit into a general framework of “fp-Mahowaldean spectra” (for “finitely presented Mahowaldean spectrum”, inspired by[MR99]), of which A , B , T ( n ), and y ( n ) would be examples. One might then hopefor a generalization of Theorem A which relates fp-Mahowaldean spectra to fp-spectra.It would also be interesting to prove an analogue of Mahowald-Rezk duality for fp-Mahowaldean spectra which recovers their duality for fp-spectra upon taking Thomspectra as above.(j) One potential approach to the question about surjectivity raised above is as follows.The surjectivity claim at height 0 is the (trivial) statement that the unit map S → H Z is surjective on homotopy. The Kahn-Priddy theorem, stating that the transfer λ : Σ ∞ R P ∞ → S is surjective on π ∗≥ , can be interpreted as stating that π ∗ Σ ∞ R P ∞ contains those elements of π ∗ S which are not detected by H Z . One is then led to wonder:for each of the Thom spectra R on the second line of Table 1, is there a spectrum P alongwith a map λ R : P → R such that each x ∈ π ∗ R in the kernel of the map R → Θ( R ) ifts along λ R to π ∗ P ? (The map R → Θ( R ) is an equivalence in dimensions < | σ n | (if R is of height n ), so P would have bottom cell in dimension | σ n | .)Since Σ ∞ R P ∞ ≃ Σ − Sym ( S ) / S , the existence of such a result is very closely tied toan analogue of the Whitehead conjecture (see [Kuh82]; the Whitehead conjecture impliesthe Kahn-Priddy theorem). In particular, one might expect the answer to the questionposed above to admit some interaction with Goodwillie calculus.(k) Let p ≥
5. Is there a p -primary analogue of B which would provide a Thom spectrumconstruction (via Table 1) of the conjectural spectrum eo p − ? Such a spectrum wouldbe the Thom spectrum of a p -complete spherical fibration over a p -local space built via p − S k ( p − for 2 ≤ k ≤ p .(l) The spectra T ( n ) and y ( n ) have algebro-geometric interpretations: the stack M T ( n ) associated (see [DFHH14, Chapter 9]; this stack is well-defined since T ( n ) is homotopycommutative) to T ( n ) classifies p -typical formal groups with a coordinate up to degree p n +1 −
1, while y ( n ) is the closed substack of M T ( n ) defined by the vanishing locus of p, v , · · · , v n − . What are the moduli problems classified by A and B ? We do not knowif this question even makes sense at p = 2, since A and B are a priori only E -rings.Nonetheless, in [Dev19a], we provide a description of tmf ∧ A in terms of the Hodgefiltration of the universal elliptic curve (even at p = 2).(m) Theorem A shows that the Hopkins-Mahowald theorem for H Z p can be generalized todescribe forms of BP h n i ; at least for small n , these spectra have associated algebro-geometric interpretations (see [DFHH14, Dev18]). What is the algebro-geometric inter-pretation of Theorem A? References [AB19] O. Antol´ın-Camarena and T. Barthel. A simple universal property of Thom ring spectra.
J. Topol. ,12(1):56–78, 2019. (Cited on pages 6 and 7.)[ABG +
14] M. Ando, A. Blumberg, D. Gepner, M. Hopkins, and C. Rezk. An ∞ -categorical approach to R -linebundles, R -module Thom spectra, and twisted R -homology. J. Topol. , 7(3):869–893, 2014. (Cited onpage 6.)[ABP67] D. Anderson, E. Brown, and F. Peterson. The structure of the Spin cobordism ring.
Ann. of Math.(2) , 86:271–298, 1967. (Cited on pages 4 and 31.)[Ada60] J. F. Adams. On the non-existence of elements of Hopf invariant one.
Ann. of Math. (2) , 72:20–104,1960. (Cited on page 10.)[AHL10] V. Angeltveit, M. Hill, and T. Lawson. Topological Hochschild homology of ℓ and ko . Amer. J. Math. ,132(2):297–330, 2010. (Cited on page 39.)[AHR10] M. Ando, M. Hopkins, and C. Rezk. Multiplicative orientations of KO -theory and of the spectrumof topological modular forms. , May 2010.(Cited on page 1.)[AL17] V. Angeltveit and J. Lind. Uniqueness of BP h n i . J. Homotopy Relat. Struct. , 12(1):17–30, 2017. (Citedon pages 4 and 27.)[AM99] G. Arone and M. Mahowald. The Goodwillie tower of the identity functor and the unstable periodichomotopy of spheres.
Invent. Math. , 135(3):743–788, 1999. (Cited on page 44.)[Ame18] S. Amelotte. A homotopy decomposition of the fibre of the squaring map on Ω S . Homology Homo-topy Appl. , 20(1):141–154, 2018. (Cited on page 37.)[Ame19] S. Amelotte. The fibre of the degree 3 map, Anick spaces and the double suspension. https://arxiv.org/abs/1908.05302 , 2019. (Cited on page 21.)[Ani93] D. Anick.
Differential algebras in topology , volume 3 of
Research Notes in Mathematics . A K Peters,Ltd., Wellesley, MA, 1993. (Cited on page 21.)[AP76] F. Adams and S. Priddy. Uniqueness of B SO.
Math. Proc. Cambridge Philos. Soc. , 80(3):475–509,1976. (Cited on pages 4 and 27.)[AQ19] G. Angelini-Knoll and J. D. Quigley. Chromatic Complexity of the Algebraic K-theory of y ( n ). https://arxiv.org/abs/1908.09164 , 2019. (Cited on page 2.)[Aro98] G. Arone. Iterates of the suspension map and Mitchell’s finite spectra with A k -free cohomology. Math.Res. Lett. , 5(4):485–496, 1998. (Cited on page 44.) Ast97] L. Astey. Commutative 2-local ring spectra.
Proc. Roy. Soc. Edinburgh Sect. A , 127(1):1–10, 1997.(Cited on page 29.)[BBB +
19] A. Beaudry, M. Behrens, P. Bhattacharya, D. Culver, and Z. Xu. On the tmf-resolution of Z. https://arxiv.org/abs/1909.13379 , 2019. (Cited on pages 35 and 44.)[BE16] P. Bhattacharya and P. Egger. A class of 2-local finite spectra which admit a v -self-map. https://arxiv.org/abs/1608.06250 , 2016. (Cited on page 35.)[Bea17] J. Beardsley. Relative Thom spectra via operadic Kan extensions. Algebr. Geom. Topol. , 17(2):1151–1162, 2017. (Cited on page 6.)[Bea18] J. Beardsley. A Theorem on Multiplicative Cell Attachments with an Application to Ravenel’s X ( n )Spectra. https://arxiv.org/abs/1708.03042v6 , 2018. (Cited on page 11.)[BMMS86] R. Bruner, P. May, J. McClure, and M. Steinberger. H ∞ ring spectra and their applications , volume1176 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, 1986. (Cited on page 8.)[BT13] P. Beben and S. Theriault. The Kahn-Priddy theorem and the homotopy of the three-sphere.
Proc.Amer. Math. Soc. , 141(2):711–723, 2013. (Cited on page 22.)[BW18] M. Behrens and D. Wilson. A C -equivariant analog of Mahowald’s Thom spectrum theorem. Proc.Amer. Math. Soc. , 146(11):5003–5012, 2018. (Cited on page 42.)[Cha19] H. Chatham. Thom Complexes and the Spectrum tmf. https://arxiv.org/abs/1903.07116 , 2019.(Cited on pages 2 and 9.)[CMN79a] F. Cohen, J. Moore, and J. Neisendorfer. The double suspension and exponents of the homotopygroups of spheres.
Ann. of Math. (2) , 110(3):549–565, 1979. (Cited on pages 3, 4, and 20.)[CMN79b] F. Cohen, J. Moore, and J. Neisendorfer. Torsion in homotopy groups.
Ann. of Math. (2) , 109(1):121–168, 1979. (Cited on pages 3, 4, and 20.)[Coh87] F. Cohen. A course in some aspects of classical homotopy theory. In
Algebraic topology (Seattle, Wash.,1985) , volume 1286 of
Lecture Notes in Math. , pages 1–92. Springer, Berlin, 1987. (Cited on page 22.)[Dev18] S. Devalapurkar. An equivariant version of Wood’s theorem. ,2018. (Cited on page 45.)[Dev19a] S. Devalapurkar. Hodge theory for elliptic curves and the Hopf element ν . https://arxiv.org/abs/1912.02548 , 2019. (Cited on page 45.)[Dev19b] S. Devalapurkar. The Ando-Hopkins-Rezk orientation is surjective. https://arxiv.org/abs/1911.10534 , 2019. (Cited on pages 2, 3, 10, 13, 15, 17, 31, 32, and 44.)[DFHH14] C. Douglas, J. Francis, A. Henriques, and M. Hill. Topological Modular Forms , volume 201 of
Math-ematical Surveys and Monographs . American Mathematical Society, 2014. (Cited on page 45.)[DHS88] E. Devinatz, M. Hopkins, and J. Smith. Nilpotence and stable homotopy theory. I.
Ann. of Math. (2) ,128(2):207–241, 1988. (Cited on pages 1, 2, 10, 30, and 43.)[DM81] D. Davis and M. Mahowald. v - and v -periodicity in stable homotopy theory. Amer. J. Math. ,103(4):615–659, 1981. (Cited on pages 15 and 34.)[DM10] D. Davis and M. Mahowald. Connective versions of
T MF (3).
Int. J. Mod. Math. , 5(3):223–252, 2010.(Cited on page 44.)[Fra13] J. Francis. The tangent complex and Hochschild cohomology of E n -rings. Compos. Math. , 149(3):430–480, 2013. (Cited on page 18.)[Gra88] B. Gray. On the iterated suspension.
Topology , 27(3):301–310, 1988. (Cited on pages 3, 4, and 21.)[Gra89a] B. Gray. Homotopy commutativity and the
EHP sequence. In
Algebraic topology (Evanston, IL,1988) , volume 96 of
Contemp. Math. , pages 181–188. Amer. Math. Soc., Providence, RI, 1989. (Citedon page 14.)[Gra89b] B. Gray. On the double suspension. In
Algebraic topology (Arcata, CA, 1986) , volume 1370 of
LectureNotes in Math. , pages 150–162. Springer, Berlin, 1989. (Cited on pages 3, 4, and 21.)[GT10] B. Gray and S. Theriault. An elementary construction of Anick’s fibration.
Geom. Topol. , 14(1):243–275, 2010. (Cited on page 21.)[HH92] M. Hopkins and M. Hovey. Spin cobordism determines real K -theory. Math. Z. , 210(2):181–196, 1992.(Cited on page 33.)[HK01] P. Hu and I. Kriz. Real-oriented homotopy theory and an analogue of the Adams-Novikov spectralsequence.
Topology , 40(2):317–399, 2001. (Cited on page 42.)[HM02] M. Hopkins and M. Mahowald. The structure of 24 dimensional manifolds having normal bundleswhich lift to B O[8]. In
Recent progress in homotopy theory (Baltimore, MD, 2000) , volume 293 of
Contemp. Math. , pages 89–110. Amer. Math. Soc., Providence, RI, 2002. (Cited on page 17.)[Hop84] M. Hopkins.
Stable decompositions of certain loop spaces . ProQuest LLC, Ann Arbor, MI, 1984. Thesis(Ph.D.)–Northwestern University. (Cited on pages 11, 38, and 43.) Hop87] M. Hopkins. Global methods in homotopy theory. In
Homotopy theory (Durham, 1985) , volume 117 of
London Math. Soc. Lecture Note Ser. , pages 73–96. Cambridge Univ. Press, Cambridge, 1987. (Citedon page 30.)[HS98] M. Hopkins and J. Smith. Nilpotence and stable homotopy theory. II.
Ann. of Math. (2) , 148(1):1–49,1998. (Cited on pages 10 and 43.)[HW19] J. Hahn and D. Wilson. Eilenberg-MacLane spectra as equivariant Thom spectra. https://arxiv.org/abs/1804.05292 , 2019. (Cited on pages 40 and 42.)[Kla18] I. Klang. The factorization theory of Thom spectra and twisted non-abelian Poincare duality.
Algebr.Geom. Topol. , 18(5):2541–2592, 2018. (Cited on page 19.)[Kuh82] N. Kuhn. A Kahn-Priddy sequence and a conjecture of G. W. Whitehead.
Math. Proc. Camb. Phil. ,92:467–483, 1982. (Cited on page 45.)[Lau04] G. Laures. K (1)-local topological modular forms. Invent. Math. , 157(2):371–403, 2004. (Cited onpages 1 and 33.)[Law19] T. Lawson. E n -ring spectra and Dyer–Lashof operations. ,2019. (Cited on pages 2, 8, 9, and 10.)[Lew78] G. Lewis. The stable category and generalized Thom spectra . ProQuest LLC, Ann Arbor, MI, 1978.Thesis (Ph.D.)–The University of Chicago. (Cited on page 9.)[Liu62] A. Liulevicius. The factorization of cyclic reduced powers by secondary cohomology operations.
Mem.Amer. Math. Soc. No. , 42:112, 1962. (Cited on page 10.)[LN14] T. Lawson and N. Naumann. Strictly commutative realizations of diagrams over the Steenrod algebraand topological modular forms at the prime 2.
Int. Math. Res. Not. IMRN , 10:2773–2813, 2014. (Citedon pages 9, 26, and 31.)[LS19] G. Laures and B. Schuster. Towards a splitting of the K (2)-local string bordism spectrum. Proc. Amer.Math. Soc. , 147(1):399–410, 2019. (Cited on page 1.)[Lur16] J. Lurie. Higher Algebra, 2016. (Cited on page 18.)[Mah79] M. Mahowald. Ring spectra which are Thom complexes.
Duke Math. J. , 46(3):549–559, 1979. (Citedon pages 2, 7, 15, and 34.)[Mah81a] M. Mahowald. b o-resolutions. Pacific J. Math. , 92(2):365–383, 1981. (Cited on pages 15, 35, and 44.)[Mah81b] M. Mahowald. The primary v -periodic family. Math. Z. , 177(3):381–393, 1981. (Cited on page 15.)[Mah82] M. Mahowald. The image of J in the EHP sequence.
Ann. of Math. (2) , 116(1):65–112, 1982. (Citedon pages 15, 35, and 44.)[Mah87] M. Mahowald. Thom complexes and the spectra b o and b u. In Algebraic topology (Seattle, Wash.,1985) , volume 1286 of
Lecture Notes in Math. , pages 293–297. Springer, Berlin, 1987. (Cited onpages 2, 9, and 25.)[Mat16] A. Mathew. The homology of tmf.
Homology Homotopy Appl. , 18(2):1–20, 2016. (Cited on pages 4,33, and 35.)[Mil81] H. Miller. On relations between Adams spectral sequences, with an application to the stable homotopyof a Moore space.
J. Pure Appl. Algebra , 20(3):287–312, 1981. (Cited on page 35.)[MNN15] A. Mathew, N. Naumann, and J. Noel. On a nilpotence conjecture of J. P. May.
J. Topol. , 8(4):917–932,2015. (Cited on page 7.)[MR99] M. Mahowald and C. Rezk. Brown-Comenetz duality and the Adams spectral sequence.
Amer. J.Math. , 121(6):1153–1177, 1999. (Cited on pages 32 and 44.)[MR09] M. Mahowald and C. Rezk. Topological modular forms of level 3.
Pure Appl. Math. Q. , 5(2, SpecialIssue: In honor of Friedrich Hirzebruch. Part 1):853–872, 2009. (Cited on page 1.)[MRS01] M. Mahowald, D. Ravenel, and P. Shick. The triple loop space approach to the telescope conjecture. In
Homotopy methods in algebraic topology (Boulder, CO, 1999) , volume 271 of
Contemp. Math. , pages217–284. Amer. Math. Soc., Providence, RI, 2001. (Cited on pages 2, 7, 13, and 35.)[MT94] M. Mahowald and R. Thompson. The fiber of the secondary suspension map.
Amer. J. Math. ,116(1):179–205, 1994. (Cited on pages 35 and 44.)[MU77] M. Mahowald and A. Unell. Bott periodicity at the prime 2 in the unstable homotopy of spheres. , 1977. (Cited on page 15.)[Nei81] J. Neisendorfer. 3-primary exponents.
Math. Proc. Cambridge Philos. Soc. , 90(1):63–83, 1981. (Citedon pages 3, 4, and 20.)[PR84] A. Pazhitnov and Y. Rudyak. Commutative ring spectra of characteristic 2.
Mat. Sb. (N.S.) ,124(166)(4):486–494, 1984. (Cited on page 9.)[PS86] A. Pressley and G. Segal.
Loop groups . Oxford Mathematical Monographs. The Clarendon Press,Oxford University Press, New York, 1986. Oxford Science Publications. (Cited on page 19.)[Rav84] D. Ravenel. Localization with respect to certain periodic homology theories.
Amer. J. Math. ,106(2):351–414, 1984. (Cited on page 10.) Rav86] D. Ravenel.
Complex cobordism and stable homotopy groups of spheres . Academic Press, 1986. (Citedon pages 11 and 14.)[Rav02] D. Ravenel. The method of infinite descent in stable homotopy theory. I. In
Recent progress in homotopytheory (Baltimore, MD, 2000) , volume 293 of
Contemp. Math. , pages 251–284. Amer. Math. Soc.,Providence, RI, 2002. (Cited on page 13.)[Rud98] Y. Rudyak. The spectra k and kO are not Thom spectra. In Group representations: cohomology, groupactions and topology (Seattle, WA, 1996) , volume 63 of
Proc. Sympos. Pure Math. , pages 475–483.Amer. Math. Soc., Providence, RI, 1998. (Cited on pages 2 and 9.)[Sel77] P. Selick.
Odd primary torsion in the homotopy groups of spheres . ProQuest LLC, Ann Arbor, MI,1977. Thesis (Ph.D.)–Princeton University. (Cited on pages 4 and 20.)[ST19] P. Selick and S. Theriault. New perspectives on the classifying space of the fibre of the double suspen-sion.
Proc. Amer. Math. Soc. , 147(3):1325–1333, 2019. (Cited on pages 20 and 21.)[Tho54] R. Thom. Quelques propri´et´es globales des vari´et´es diff´erentiables.
Comment. Math. Helv. , 28:17–86,1954. (Cited on page 2.)[Tod62] H. Toda.
Composition methods in homotopy groups of spheres . Annals of Mathematics Studies, No.49. Princeton University Press, Princeton, N.J., 1962. (Cited on page 21.)[Wur86] U. Wurgler. Commutative ring-spectra of characteristic 2.
Comment. Math. Helv. , 61(1):33–45, 1986.(Cited on page 9.)[Zhu17] X. Zhu. An introduction to affine Grassmannians and the geometric Satake equivalence. In
Geometryof moduli spaces and representation theory , volume 24 of
IAS/Park City Math. Ser. , pages 59–154.Amer. Math. Soc., Providence, RI, 2017. (Cited on page 19.)
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