Higher chromatic Thom spectra via unstable homotopy theory

We investigate implications of an old conjecture in unstable homotopy theory related to the Cohen-Moore-Neisendorfer theorem and a conjecture about the $\mathbf{E}_{2}$-topological Hochschild 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 $\mathrm{MSpin}\to \mathrm{ko}$ and $\mathrm{MString}\to \mathrm{tmf}$ admit spectrum-level splittings. This is shown by generalizing a theorem of Hopkins and Mahowald, which constructs $\mathrm{H}\mathbf{F}_p$ as a Thom spectrum, to construct $\mathrm{BP}\langle{n-1}\rangle$, $\mathrm{ko}$, and $\mathrm{tmf}$ as Thom spectra (albeit over $T(n)$, $A$, and $B$ respectively, and not over the sphere). This interpretation of $\mathrm{BP}\langle{n-1}\rangle$, $\mathrm{ko}$, and $\mathrm{tmf}$ offers a new perspective on Wood equivalences of the form $\mathrm{bo} \wedge C\eta \simeq \mathrm{bu}$: they are related to the existence of certain EHP sequences in unstable homotopy theory. This construction of $\mathrm{BP}\langle{n-1}\rangle$ also provides a different lens on the nilpotence theorem. Finally, we prove a $C_2$-equivariant analogue of our construction, describing $\underline{\mathrm{H}\mathbf{Z}}$ as a Thom spectrum.

1. Introduction 1.1. Statement of the main results. One of the goals of this article is to describe a program to prove the following old conjecture (studied, for instance, in [Lau04,LS19], and discussed informally in many places, such as [MR09, Section 7]): Conjecture 1.1.1. The Ando-Hopkins-Rezk orientation (see [AHR10]) MString → tmf admits a spectrum-level splitting.
The key idea in our program is to provide a universal property for mapping out of the spectrum tmf. We give a proof which is conditional on an old conjecture from unstable homotopy theory stemming from the Cohen-Moore-Neisendorfer theorem and a conjecture about the E 2 -topological Hochschild cohomology of certain Thom spectra (the latter of which simplifies the proof of the nilpotence theorem from [DHS88]). This universal property exhibits tmf as a certain Thom spectrum, similarly to the Hopkins-Mahowald construction of HZ p and HF p as Thom spectra.
To 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 HF 2 . The simplest way to do so is to show that MO is an HF 2module, which in turn can be done by constructing an E 2 -map HF 2 → MO. The construction of such a map is supplied by the following theorem of Hopkins and Mahowald: Theorem (Hopkins-Mahowald; see [Mah79] and [MRS01,Lemma 3.3]). Let µ : Ω 2 S 3 → BO denote the real vector bundle over Ω 2 S 3 induced by extending the map S 1 → BO classifying the Möbius bundle. Then the Thom spectrum of µ is equivalent to HF 2 as an E 2 -algebra.
Our argument for Conjecture 1.1.1 takes this approach: we shall show that an old conjecture from unstable homotopy theory and a conjecture about the E 2topological Hochschild cohomology of certain Thom spectra provide a construction of tmf (as well as bo and BP n ) as a Thom spectrum, and utilize the resulting universal property of tmf to construct an (unstructured) map tmf → MString.
Mahowald was the first to consider the question of constructing spectra like bo and tmf as Thom spectra (see [Mah87]). Later work by Rudyak in [Rud98] sharpened Mahowald's results to show that bo and bu cannot appear as the Thom spectrum of a p-complete spherical fibration. In [AHL09], Angeltveit-Hill-Lawson gave an alternative proof of this fact under the assumption that the p-complete spherical fibration is classified by a map of E 3 -spaces. Recently, Chatham has shown in [Cha19] that tmf ∧ 2 cannot appear as the Thom spectrum of a structured 2-complete spherical fibration over a loop space. Our goal is to argue that these issues are alleviated if we replace "spherical fibrations" with "bundles of R-lines" for certain well-behaved spectra R.
The first hint of tmf being a generalized Thom spectrum comes from a conjecture of Hopkins and Hahn regarding a construction of the truncated Brown-Peterson spectra BP n as Thom spectra. 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 a prime p, the spectra X(k) for p n ≤ k ≤ p n+1 − 1 split as a direct sum of suspensions of certain homotopy commutative ring spectra T (n), which in turn filter the gap between the p-complete sphere spectrum and BP (in the sense that T (0) = S and T (∞) = BP). Then: Conjecture 1.1.3 (Hahn, Hopkins; unpublished). There is a map f : Ω 2 S |vn|+3 → BGL 1 (T (n)), which detects an indecomposable element v n ∈ π |vn| T (n) on the bottom cell of the source, whose Thom spectrum is a form of BP n − 1 .
The primary obstruction to proving that a map f as in Conjecture 1.1.3 exists stems from the failure of T (n) to be an E 3 -ring (due to Lawson; [Law20, Example 1.5.31]). If R is an E 1 -or E 2 -ring spectrum, let Z 3 (R) denote the E 2 -topological Hochschild cohomology of R (see Definition 3.3.2). Hahn suggested that one way to get past the failure of T (n) to be an E 3 -ring would be via the following conjecture: Conjecture 1.1.4 (Hahn). There is an indecomposable element v n ∈ π |vn| T (n) which lifts to the E 2 -topological Hochschild cohomology Z 3 (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 is true). We shall instead show that Conjecture 1.1.3 is implied by the two conjectures alluded to above. We shall momentarily state these conjectures precisely as Conjectures D and E; let us first 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 ΩJ p n −1 (S 2 ) (resp. ΩJ p n −1 (S 2 ) 2 ) introduced in [MRS01] to study the telescope conjecture (resp. in [AQ19] as z(n)). Let A denote the E 1 -quotient S/ /ν of the sphere spectrum by ν ∈ π 3 (S); its mod 2 homology is H * (A) ∼ = F 2 [ζ 4 1 ]. The spectrum A has been intensely studied by Mahowald and his coauthors in (for instance) [Mah79,DM81,Mah81b,Mah81a,Mah82,MU77], where it is often denoted X 5 . (See Remark 2.1.8 for motivation for the term "E 1 -quotient".) Let B denote the E 1 -ring introduced in [Dev19b, Construction 3.1]; it has been briefly studied under the name X in [HM02]. It may be constructed as the Thom spectrum of a vector bundle over an E 1 -space N which sits in a fiber sequence ΩS 9 → N → ΩS 13 . The mod 2 homology of B is H * (B) ∼ = F 2 [ζ 8 1 , ζ 4 2 ]. We also need to recall some unstable homotopy theory. In [CMN79a,CMN79b,Nei81], Cohen, Moore, and Neisendorfer constructed a map φ n : Ω 2 S 2n+1 → S 2n−1 whose composite with the double suspension E 2 : S 2n−1 → Ω 2 S 2n+1 is the degree p map. (The symbol E stands for "Einhängung", which is German for "suspension".) Such a map was also constructed by Gray 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 2 S 2p n+1 +1 → S 2p n+1 −1 . Then Conjectures D and E imply that there is a map µ : K n+1 → BGL 1 (R) such that the mod p homology of the Thom spectrum K µ n+1 is isomorphic to the mod p homology of the associated designer chromatic spectrum Θ(R) as a Steenrod comodule 1 .
If R is any base spectrum other than B, the Thom spectrum K µ n+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 Thom spectrum K µ n+1 is equivalent to Θ(B) = tmf upon 2-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) HZ p bo tmf BP n k(n) k Z (n) Table 1. To go from a base spectrum "of height n", say R, in the second line to the third, one takes the Thom spectrum of a bundle of R-lines over K n+1 .
Making sense of Theorem A relies on knowing that T (n) admits the structure of an E 1 -ring; this is proved in [BL21]; see also Warning 3.1.6. Note that the spectra A, B, y(n), and y Z (n) all admit E 1 -structures by construction. In Remark 5.4.7, we sketch how Theorem A relates to the proof of the nilpotence theorem.
Although the form of Theorem A does not resemble Conjecture 1.1.3, we show that Theorem A implies the following result.
Corollary B. Conjectures D and E imply Conjecture 1.1.3.
In the case n = 0, Corollary B recovers the Hopkins-Mahowald theorem constructing HF p . Moreover, Corollary B is true unconditionally when n = 0, 1.
Using the resulting universal property of tmf, we obtain a result pertaining to Conjecture 1.1.1.
Theorem C. Assume that the composite Z 3 (B) → B → MString is an E 3 -map. Then Conjectures D, E, and F imply that there is a spectrum-level unital splitting of the Ando-Hopkins-Rezk orientation MString (2) → tmf (2) .
In particular, Conjecture 1.1.1 follows (at least after localizing at p = 2; a slight modification of our arguments should work at any prime). We believe that the assumption that the composite Z 3 (B) → B → MString is an E 3 -map is too strong: we believe that it can be removed using special 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 refined to structured splittings. In [Dev19b], we showed (unconditionally) that the Ando-Hopkins-Rezk orientation MString → tmf induces a surjection on homotopy, a result which is clearly implied by Theorem C.
We remark that the argument used to prove Theorem C shows that if the composite Z 3 (A) → A → MSpin is an E 3 -map, then Conjectures D and E imply that there is a spectrum-level unital splitting of the Atiyah-Bott-Shapiro orientation MSpin → bo. This splitting was originally proved 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 second part of this conjecture is a compilation of several old conjectures in unstable homotopy theory originally 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 statements made in the literature: for instance, in Conjecture D(b), we demand a Q 1 -space splitting (Notation 2.2.6), rather than merely an 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 space to the loop space on an Anick space (Definition 4.1.3). (b) There exists a p-local charming map f : Ω 2 S 2p n +1 → S 2p n −1 whose homotopy fiber admits a Q 1 -space retraction off of Ω 2 (S 2p n /p). There are also integrally defined maps Ω 2 S 9 → S 7 and Ω 2 S 17 → S 15 whose composites with the double suspension on S 7 and S 15 respectively are the degree 2 maps. Moreover, their homotopy fibers K 2 and K 3 (respectively) admit deloopings, and admit Q 1 -space retractions off Ω 2 (S 8 /2) and Ω 2 (S 16 /2) (respectively).
Next, we turn to Conjecture E. This conjecture is concerned with the E 2topological Hochschild cohomology of the Thom spectra X(p n − 1) (p) , A, and B introduced above.
Conjecture E. Let n ≥ 0 be an integer. Let R denote X(p n+1 − 1) (p) , A (in which case n = 1), or B (in which case n = 2). Then the element σ n ∈ π |vn|−1 R lifts to the E 2 -topological Hochschild cohomology Z 3 (R) of R, and is p- Finally, we state Conjecture F. It is inspired by [AP76,AL17]. We believe this conjecture is the most approachable of the conjectures stated here.
Conjecture F. Suppose X is a spectrum which is bounded below and whose homotopy groups are finitely generated over Z p . If there is an isomorphism H * (X; F p ) ∼ = H * (tmf; F p ) of Steenrod comodules, then there is a homotopy equivalence X ∧ p → tmf ∧ p of spectra. After proving Theorem A and Theorem C, we explore relationships between the different spectra appearing on the second line of Table 1 in the remainder of the article. In particular, we prove analogues of Wood's equivalence bo ∧ Cη ≃ bu (see also [Mat16]) for these spectra. We argue that these are related to the existence of certain EHP sequences.
Finally, we describe a C 2 -equivariant analogue of Corollary B at n = 1 as Theorem 7.2.1, independently of a C 2 -equivariant analogue of Conjecture D and Conjecture E. This result constructs HZ as a Thom spectrum of an equivariant bundle of invertible T (1) R -modules over Ω ρ S 2ρ+1 , where T (1) R is the free E σalgebra with a nullhomotopy of the equivariant Hopf map η ∈ π σ (S), and ρ and σ are the regular and sign representations of C 2 , respectively. This uses results of Behrens-Wilson and Hahn-Wilson from [BW18,HW20]. We believe there is a similar result at odd primes, but we defer discussion of this. We discuss why our methods do not work to yield BP n R for n ≥ 1 as in Corollary B.
1.3. Outline. Section 2 contains a review some of the theory of Thom spectra from the modern perspective, as well as the proof of the classical Hopkins-Mahowald theorem. The content reviewed in this section will appear in various guises throughout this project, hence its inclusion.
In Section 3, we study certain E 1 -rings; most of them appear as Thom spectra over the sphere. For instance, we recall some facts about Ravenel's X(n) spectra, and then define and prove properties about the E 1 -rings A and B used in the statement of Theorem A. We state Conjecture 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-Neisendorfer map and the fiber of the double suspension. These concepts do not show up often in stable homotopy theory, so we hope this section provides useful background to the reader. We state Conjecture 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 consequences of Theorem A.
In Section 6, we study some applications of Theorem A. For instance, we use it to prove Theorem C, which is concerned with the splitting of certain cobordism spectra. In a previous version of this article, we had two subsections discussing Wood-like equivalences, and topological Hochschild homology of the chromatic Thom spectra of Table 1. However, while making revisions to this article, we decided to split these two sections off into separate articles [Dev22a,Dev22b].
In Section 7, we prove an equivariant analogue of Corollary B at height 1. We construct equivariant analogues of X(n) and A, and describe why our methods fail to produce an equivariant analogue of Corollary B at all heights, even granting an analogue of Conjecture D and Conjecture E.
Finally, in Section 8, we suggest some directions for future research. There are also numerous interesting questions arising from our work, which we have indicated in the body of the article.
Conventions. Unless indicated otherwise, or if it goes against conventional notational choices, a Latin letter with a numerical subscript (such as x 5 ) denotes an element of degree given by its subscript. If X is a space and R is an E 1 -ring spectrum, then X µ will denote the Thom spectrum of some bundle of invertible Rmodules determined by a map µ : X → BGL 1 (R). We shall often quietly localize or complete at an implicit prime p. Although we have tried to be careful, all limits and colimits will be homotopy limits and colimits; we apologize for any inconvenience this might cause.
We shall denote by P k (p) the mod p Moore space S k−1 ∪ 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 written nowadays as ξ i and τ i , although, as Haynes Miller pointed out to me, our notation for the conjugates was Milnor's original notation) in degrees 2(p i − 1) and 2p i − 1 for p > 2 and 2 i − 1 (for ζ i ) at p = 2. Unfortunately, we will use A to denote the E 1 -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. A map of O-rings respecting the O-algebra structure will often simply be called a O-map. Unless it is clear that we mean otherwise, all modules over non-E ∞ -algebras will be left modules.
Hood Chatham pointed out to me that S 3 4 would be the correct notation for what we denote by S 3 3 = fib(S 3 → K(Z, 3)). Unfortunately, the literature seems to have chosen S 3 3 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 mean that 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.
Acknowledgements. The genesis of this project took place in conversations with Jeremy Hahn, who has been a great friend and mentor, and a fabulous resource; I'm glad to be able to acknowledge our numerous discussions, as well as his comments on a draft of this article. I'm extremely grateful to Mark Behrens and Peter May for working with me over the summer of 2019 and for being fantastic advisors, as well as for arranging my stay at UChicago, where part of 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 project also took place at MIT and Boulder, and I'd like to thank Araminta Gwynne, Robert Burklund, Hood Chatham, Peter Haine, Mike Hopkins, Tyler Lawson, Kiran Luecke, Andrew Senger, Neil Strickland, and Dylan Wilson for clarifying discussions. Although I never got the chance to meet Mark Mahowald, my intellectual debt to him is hopefully evident (simply Ctrl+F his name in this document!). I would like to the anonymous referee for several comments which greatly improved this article. Finally, I'm glad I had the opportunity to meet other math nerds at the UChicago REU; I'm in particular calling out Ada, Anshul, Eleanor, and Wyatt -thanks for making my summer enjoyable.
Part of this work was done when the author was supported by the PD Soros Fellowship.
2. Background, and some classical positive and negative results 2.1. Background on Thom spectra. In this section, we will recall some facts about Thom spectra and their universal properties; the discussion is motivated by [ABG + 14].
Definition 2.1.1. Let A be an E 1 -ring, and let µ : X → BGL 1 (A) be a map of spaces. The Thom A-module X µ is defined as the homotopy pushout Remark 2.1.2. Let A be an E 1 -ring, and let µ : X → BGL 1 (A) be a map of spaces. The Thom A-module X µ is the homotopy colimit of the functor X µ − → BGL 1 (A) → Mod(A), where we have abused notation by identifying X with its associated Kan complex. If A is an E 1 -R-algebra, then the R-module underlying X can be identified with the homotopy colimit of the composite functor where we have identified X with its associated Kan complex. The space B Aut R (A) can be regarded 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 1 -rings with an E 1 -ring map R → R ′ exhibiting R ′ as a right R-module. If f : X → BGL 1 (R) is a map of spaces, then the Thom spectrum of the composite Corollary 2.1.4. Let R and R ′ be E 1 -rings with an E 1 -ring map R → R ′ exhibiting R ′ as a right R-module. If f : X → BGL 1 (R) is a map of spaces such that the the composite X → BGL 1 (R) → BGL 1 (R ′ ) is null, then there is an equivalence 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 Thom spectrum of an E k -map X → BGL 1 (R) is an E k -R-algebra.
We will repeatedly use the following classical result, which is again a consequence of the observation that Thom spectra are colimits, as well as the fact that total spaces of fibrations may be expressed as colimits; see also [Bea17, Theorem 1].
Proposition 2.1.6. Let X i − → Y → Z be a fiber sequence of k-fold loop spaces (where k ≥ 1), and let R be an E m -ring for m ≥ k + 1. Suppose that µ : Y → BGL 1 (R) is a map of k-fold loop spaces. Then, there is a k-fold loop map φ : Z → BGL 1 (X µ•i ) whose Thom spectrum is equivalent to Y µ as E k−1 -rings. Concisely, if arrows are labeled by their associated Thom spectra, then there is a diagram The argument to prove Proposition 2.1.6 also goes through with slight modifications when k = 0, and shows: be a fiber sequence of spaces with Z connected, and let R be an E m -ring for m ≥ 1. Suppose that µ : Y → BGL 1 (R) is a map of Kan complexes. Then, there is a map φ : Z → B Aut R (X µ•i ) such that the homotopy colimit (i.e., "Thom spectrum") Z φ of the following composite 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.
Proof of the second form of Proposition 2.1.6. It will be convenient to use the model for Thom spectra following [ABG + 14]. Observe that a fibration X → Y → Z implies (e.g., by [ABG + 14, Remark 2.4]) that there is a functor Z → Top whose homotopy colimit is Y , and whose fiber over any vertex of z ∈ Z is X. Since X is connected, we may write Y ≃ hocolim Z X. The map X → Y is induced by the inclusion {z} ֒→ Z. Since Y is a Kan complex, the Thom spectrum Y µ can be identified (by [ABG + 14, Definition 1.4]) with the homotopy colimit of the composite Y µ − → BGL 1 (R) ≃ R-line ⊆ Mod R (which we will temporarily denote by µ : Y → Mod R ). We will write this as Y µ ≃ hocolim Y R. The left Kan extension of the map Y → Z along the functor µ : Y → Mod R defines a functor φ : Z → Mod R , which sends z ∈ Z to X µ•i ≃ hocolim(X → Y µ − → Mod R ). Since Z is connected, this implies that Y µ ≃ hocolim Y R is the homotopy colimit of the functor (2.1).
The following is a slight generalization of [AB19, Theorem 4.10]: Theorem 2.1.7. Let R be an E k+1 -ring for k ≥ 0, and let α : Remark 2.1.8. Say Y = S n+1 , so α detects an element α ∈ π n R. Theorem 2.1.7 suggests interpreting the Thom spectrum (Ω m S m+n+1 ) α as an E m -quotient; to signify this, we will denote it by R/ / Em 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. See [AB19, Definition 4.3], where the quotient R/ / Em α is called the versal R-algebra of characteristic α.

The
Hopkins-Mahowald theorem. The primary motivation for this project is the following miracle (see [Mah79] for p = 2 and [MRS01, Lemma 3.3] for p > 2, as well as [AB19, Theorem 5.1] for a proof of the equivalence as one of E 2 -algebras): Theorem 2.2.1 (Hopkins-Mahowald). Let S ∧ p be the p-completion of the sphere at a prime p, and let f : It is not too hard to deduce the following result from Theorem 2.2.1: Corollary 2.2.2. Let S 3 3 denote the 3-connected cover of S 3 . Then the Thom spectrum of the composite Ω 2 S 3 3 → Ω 2 S 3 µ − → BGL 1 (S ∧ p ) is equivalent to HZ p as an E 2 -ring.
Remark 2.2.3. Theorem 2.2.1 implies a restrictive version of the nilpotence theorem: if R is an E 2 -ring spectrum, and x ∈ π * R is a simple p-torsion element which has trivial HF p -Hurewicz image, then x is nilpotent. This is explained in [MNN15,Proposition 4.19]. Indeed, to show that 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 2 -ring in which p = 0, so the universal property of Theorem 2.1.7 implies that there is an E 2 -map HF p → R[1/x]. It follows that the unit R → R[1/x] factors through the Hurewicz map R → R ∧ HF p . In particular, the multiplication by x map on R[1/x] factors as the indicated dotted map: HF p ∧ Σ |x| R x / / R ∧ HF p 9 9 s s s s s However, the bottom map is null (because x has trivial HF p -Hurewicz image), so x must be null in π * R[1/x]. This is possible if and only if R[1/x] is contractible, as desired. See Proposition 5.4.1 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 in Step 2 of Section 5.2, we will recall a proof of this theorem. The key non-formal input is the following result of Steinberger's from [BMMS86, Theorems III.2.2 and III.2.3]: Theorem 2.2.4 (Steinberger). Let ζ i denote the conjugate to the Milnor generators ξ i of the dual Steenrod algebra, and similarly for τ i at odd primes. Then Proof of Theorem 2.2.1. By Corollary 2.1.7, the Thom spectrum (Ω 2 S 3 ) µ is the free E 2 -ring with a nullhomotopy of p. Since HF p is an E 2 -ring with a nullhomotopy of p, we obtain an E 2 -map (Ω 2 S 3 ) µ → HF p . To prove that this map is a p-complete equivalence, it suffices to prove that it induces an isomorphism on mod p homology.
The mod p homology of (Ω 2 S 3 ) µ can be calculated directly via the Thom isomorphism HF p ∧(Ω 2 S 3 ) µ ≃ HF p ∧Σ ∞ + Ω 2 S 3 . Note that this is not an equivalence as HF p ∧ HF p -comodules: the Thom twisting is highly nontrivial.
For simplicity, we will now specialize to the case p = 2, although the same proof works at odd primes. The homology of Ω 2 S 3 is classical: it is a polynomial ring generated by applying E 2 -Dyer-Lashof operations to a single generator x 1 in degree 1. Theorem 2.2.4 implies that the same is true for the mod 2 Steenrod algebra: it, too, is a polynomial ring generated by applying E 2 -Dyer-Lashof operations to the single generator ζ 1 = ξ 1 in degree 1. Since the map (Ω 2 S 3 ) µ → HF 2 is an E 2 -ring map, it preserves E 2 -Dyer-Lashof operations on mod p homology. By the above discussion, it suffices to show that the generator To prove this, note that x 1 is the image of the generator in degree 1 in homology under the double suspension S 1 → Ω 2 S 3 , and that ζ 1 is the image of the generator in degree 1 in homology under the canonical map S/p → HF p . It therefore suffices to show that the Thom spectrum of the spherical fibration S 1 → BGL 1 (S ∧ p ) detecting 1 − p is simply S/p. This is an easy exercise.
Remark 2.2.5. When p = 2, one does not need to p-complete in Theorem 2.2.1: the map S 1 → BGL 1 (S ∧ 2 ) factors as S 1 → BO → BGL 1 (S), where the first map detects the Möbius bundle over S 1 , and the second map is the J-homomorphism.
Notation 2.2.6. Let Q 1 denote the (operadic nerve of the) cup-1 operad from [Law20, Example 1.3.6]: this is the operad whose nth space is empty unless n = 2, in which case it is S 1 with the antipodal action of Σ 2 . We will need to slightly modify the definition of Q 1 when localized at an odd prime p: in this case, it will denote the operad whose nth space is a point if n < p, empty if n > p, and when n = p is the ordered configuration space Conf p (R 2 ) with the permutation action of Σ p . Any homotopy commutative ring admits the structure of a Q 1 -algebra at p = 2, but at other primes it is slightly stronger to be a Q 1 -algebra than to be a homotopy commutative ring. If k ≥ 2, any E k -algebra structure on a spectrum restricts to a Q 1 -algebra structure.
Remark 2.2.7. As stated in [Law20, Proposition 1.5.29], the operation Q 1 already exists in the mod 2 homology of any Q 1 -ring R, where Q 1 is the cup-1 operad from Notation 2.2.6 -the entire E 2 -structure is not necessary. With our modification of Q 1 at odd primes as in Remark 2.2.6, this is also true at odd primes.
Remark 2.2.8. We will again momentarily specialize to p = 2 for convenience. Steinberger's calculation in Theorem 2.2.4 can be rephrased as stating that Q 1 ζ i = ζ i+1 , where Q 1 is the lower-indexed Dyer-Lashof operation. (See [BMMS86, Page 59] for this notation.) As in Remark 2.2.7, the operation Q 1 already exists in the mod p homology of any Q 1 -ring R. Since homotopy commutative rings are Q 1 -algebras in spectra, this observation can be used to prove results of Würgler ([Wur86, Theorem 1.1]) and Pazhitnov-Rudyak ([PR84, Theorem in Introduction]).
Remark 2.2.9. The argument with Dyer-Lashof operations and Theorem 2.2.4 used in the proof of Theorem 2.2.1 will be referred to as the Dyer-Lashof hopping argument. It will be used again (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), as well as to Bökstedt's calculation of THH(F p ) (as a ring spectrum, and not just the calculation of its homotopy). Let us sketch an argument. First, Theorem 2.2.4 implies Theorem 2.2.1 (by the proof above). The other direction (i.e., the calculation (2.2)) can be argued by observing that the Thom isomorphism HF p ∧ HF p ≃ HF p ∧ Σ ∞ + Ω 2 S 3 is an equivalence of E 2 -HF p -algebras, so that the Dyer-Lashof operations are determined by the operations in H * (Ω 2 S 3 ; F p ). But the Dyer-Lashof operations are defined by classes in H * (Ω 2 S 3 ; F p ), and Theorem 2.2.4 is a consequence of the fact that the iterates of Q 1 on the generator of H 1 (Ω 2 S 3 ; F p ) describe all the polynomial generators H * (Ω 2 S 3 ; F p ).
It remains to argue that Theorem 2.2.1 is equivalent to the calculation that THH(F p ) ≃ F p [ΩS 3 ] as an E 1 -F p -algebra. This is showed in [KN19, Remark 1.5].
2.3. No-go theorems for higher chromatic heights. In light of Theorem 2.2.1 and Corollary 2.2.2, it is natural to wonder if appropriate higher chromatic analogues of HF p and HZ, such as BP n , bo, or tmf can be realized as Thom spectra of spherical fibrations. The answer is known to be negative (see [Mah87,Rud98,Cha19]) in many cases: Rudyak, Chatham). There is no space X with a spherical fibration µ : X → BGL 1 (S) (even after completion) such that X µ is equivalent to BP 1 or bo. Moreover, there is no 2-local loop space X ′ with a spherical fibration determined by an H-map µ : 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 argument used by Mahowald to show that bu is not the Thom spectrum of a spherical fibration over a loop space classified by an H-map can be used to show that BP 2 at p = 2 (i.e., tmf 1 (3)) is not the Thom spectrum of a spherical fibration over a loop space classified by an H-map. We do not know a proof that BP n is not the Thom spectrum of a spherical fibration over a loop space classified by an H-map for all n ≥ 1 and all primes, but we strongly suspect this to be true.
we find that the mod p homology of Ω 2 X would be isomorphic as an algebra to a polynomial ring on infinitely many generators, possibly tensored with an exterior algebra on infinitely many generators. The Eilenberg-Moore spectral sequence then implies that the mod p cohomology of X is given by where |b i | = 2p i and |c i | = 2p i−1 + 1. If p is odd, then since |b 1 | = 2p, we have P p (b 1 ) = b p 1 . Liulevicius' formula for P 1 in terms of secondary cohomology operations ([Liu62, Theorem 1]) allows us to write P p (b 1 ) as a sum c 0 R(b 1 ) + γ c 0,γ Γ γ (b 1 ), where R(b 1 ) is a coset in H 2p+4(p−1) (X; F p ) and Γ γ is an operation of odd degree, so that Γ γ (b 1 ) is in odd degree. We will not need to know what exactly the sum is indexed by, or what any of these operations are. Observe that Γ γ kills b 1 because everything is concentrated in even degrees in the relevant range, and R also kills b 1 since |R(b 1 )| = 4(p−1)+2p i is never a sum of numbers of the form 2p k when p > 2. Using this, one can conclude that b p 1 = 0, which is a contradiction. A similar calculation works at p = 2, using Adams' study of 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 if we replace BGL 1 (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.

Some Thom spectra
In this section, we introduce certain E 1 -rings; most of them appear as Thom spectra over the sphere. The following table summarizes the spectra introduced in this section and gives references to their locations in the text. The spectra A and B were introduced in [Dev19b].
where the first map arises from Bott periodicity.
Since the map ΩSU(n) → BU is an equivalence in dimensions ≤ 2n − 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 not detected by MU is 2n − 1.
Remark 3.1.4. The proof of the nilpotence theorem shows that each of the X(n) detects nilpotence. However, it is known (see [Rav84, Theorem 3.1]) that X(n) > X(n + 1) .
After localizing at a prime p, the spectrum MU splits as a wedge of suspensions of BP; this splitting comes from the Quillen idempotent on MU. The same is true of the X(n) spectra, as explained in [Rav86, Section 6.5]: a multiplicative map , with a 0 = 1 and a i ∈ π 2i (X(n) (p) ). One can use this to define a truncated form of the Quillen idempotent ǫ n on X(n) (p) (see [Hop84, Proposition 1.3.7]), and thereby obtain 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 − 1. Then X(k) (p) splits as a wedge of suspensions 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 | − 2, so there is an indecomposable element v i ∈ π * T (n) which maps to an indecomposable element in π * BP for 0 ≤ i ≤ n.
• T (n) is a homotopy associative and Q 1 -algebra spectrum.
Remark 3.1.6. It is known that that T (n) admits the structure of an E 1 -ring (see [BL21,Section 7.5]). We will interpret the phrase "Thom spectrum X µ of a map µ : It is believed that T (n) in fact admits more structure (see [AQ21, Section 6] for some discussion): Conjecture 3.1.7. The Q 1 -ring structure on T (n) extends to an E 2 -ring structure.
Remark 3.1.9. Conjecture 3.1.7 is true at p = 2 and n = 2. The Stiefel manifold V 2 (H 2 ) sits in a fiber sequence There is an equivalence V 2 (H 2 ) ≃ Sp(2), so ΩV 2 (H 2 ) admits the structure of a double loop space. There is an E 2 -map µ : ΩV 2 (H 2 ) → BU, given by taking double loops of the composite The map µ admits a description as the left vertical map in the following map of fiber sequences: Here, the map S 3 → B 2 U detects the generator of π 2 (BU) (which maps to η ∈ π 2 (BGL 1 (S)) under the J-homomorphism). The Thom spectrum ΩV 2 (H 2 ) µ is equivalent to T (2), and it follows that T (2) admits the structure of an E 2 -ring. We do not know whether T (n) is the Thom spectrum of a p-complete spherical fibration over some space for n ≥ 3.
According to Proposition 2.1.6, the spectrum X(n + 1) is the Thom spectrum of an E 1 -map ΩS 2n+1 → BGL 1 (ΩSU(n)) µ = BGL 1 (X(n)). This E 1 -map is the extension of a map S 2n → BGL 1 (X(n)) which detects an element χ n ∈ π 2n−1 X(n). This element is equivalently determined by the map Σ ∞ + Ω 2 S 2n+1 → X(n) given by the Thomification of the nullhomotopic composite Figure 1. Cη ∧ Cν shown horizontally, with 0-cell on the left. The element σ 1 is given by the map η on the 4-cell defined by a nullhomotopy of ην = 0 ∈ π 4 (S 0 ), as indicated in the diagram above.
where the first two maps form a fiber sequence. By Proposition 2.1.6, X(n + 1) is the free E 1 -X(n)-algebra with a nullhomotopy of χ n .
Remark 3.1.11. Another construction of the map χ n ∈ π 2n−1 X(n) from Construction 3.1.10 is as follows. There is a map i : CP n−1 → ΩSU(n) given by sending a line ℓ ⊆ C n to the loop S 1 → SU(n) = Aut(C n , , ) defined as follows: θ ∈ S 1 is sent to the (appropriate rescaling of the) 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 −2 CP n → X(n). The map χ n is then the composite where the first map is the desuspension of the generalized Hopf map S 2n+1 → CP n which attaches the top cell of CP n+1 . The fact that this map is indeed χ n follows immediately from the commutativity of the following diagram: 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.
3.2. Related Thom spectra. We now introduce several Thom spectra related to the E 1 -rings T (n) described in the previous section; some of these were introduced in [Dev19b]. (Relationships to T (n) will be further discussed in Section 6.2.) For the reader's convenience, we have included a table of the spectra introduced below with internal references to their definitions at the beginning of this section.
Definition 3.2.2. Let y(n) denote the Thom spectrum of the composite If J p n −1 (S 2 ) 2 denotes the 2-connected cover of J p n −1 (S 2 ), then let y Z (n) denote the Thom spectrum of the composite , so that both y(n) and y Z (n) admit the structure of E 1 -rings via [AB19, Corollary 3.2].
Proposition 3.2.3. As BP * BP-comodules, we have There is an equivalence y Z (n)/p ≃ y(n), so that BP * (y Z (n))/p ≃ BP * (y(n)). The Bockstein spectral sequence collapses, and the extensions on the E ∞ -page simply place p in filtration 1. This implies the second equivalence.
One corollary is the following; this can be deduced from Proposition 3.2.3 using Remark 3.2.1. We also refer to [AQ19, Lemma 2.3] for a direct proof of the following.
Corollary 3.2.4. As A * -comodules, we have We will now relate y(n) and y Z (n) to T (n).
Construction 3.2.5. Let m ≤ n, and let I m be the ideal generated by p, v 1 , · · · , v m−1 , where the v i are some choices of indecomposables in π |vi| (T (n)) which form a regular sequence. Inductively define T (n)/I m as the cofiber of the map Proposition 3.2.6. Let p > 2. There is an equivalence between T (n)/I n (resp. T (n)/(v 1 , · · · , v n−1 )) and the spectrum y(n) (resp. y Z (n)) of Definition 3.2.2.
Proof. We will prove the result for y(n); the analogous proof works for y Z (n). By [Gra89a], the space ΩJ p n −1 (S 2 ) is homotopy commutative (since p > 2). Moreover, the map ΩJ p n −1 (S 2 ) → Ω 2 S 3 is an H-map, so y(n) is a homotopy commutative E 1 -ring spectrum. It is known (see [Rav86, Section 6.5]) that homotopy commutative maps T (n) → y(n) are equivalent to partial complex orientations of y(n), i.e., factorizations Such a γ n indeed exists by obstruction theory: suppose k < p n − 1, and we have a map Σ −2 CP k → y(n). Since there is a cofiber sequence of spectra, the obstruction to extending along Σ −2 CP k+1 is an element of π 2k−1 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 compatible family in the sense that there is a commutative diagram The formal group law over HF p has infinite height; this forces the elements p, v 1 , · · · , v n−1 (defined for the "(p n − 1)-bud" on π * y(n)) to vanish in the homotopy of y(n). It follows that the orientation T (n) → y(n) constructed above factors through the quotient T (n)/I n . The induced map T (n)/I n → y(n) can be seen to be an isomorphism on homology (via, for instance, Definition 3.2.2 and Construction 3.2.5).
Remark 3.2.7. Since y(n) has a v n -self-map, we can form the spectrum y(n)/v n ; its mod p homology is It is in fact possible to give a construction of y(1)/v 1 as a spherical Thom spectrum. We will work at p = 2 for convenience. Define Q to be the fiber of the map 2η : S 3 → S 2 . There is a map of fiber sequences By [DM81, Theorem 3.7], the Thom spectrum of the leftmost map is y(1)/v 1 .
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 studied in [Dev19b].
Definition 3.2.8. Let S 4 → BSpin denote the generator of π 4 BSpin ∼ = Z, and let ΩS 5 → BSpin denote the extension of this map, which classifies a real vector bundle of virtual dimension zero over ΩS 5 . Let A denote the Thom spectrum of this bundle.
Remark 3.2.9. As mentioned in the introduction, the spectrum A has been intensely studied by Mahowald and his coauthors in (for instance) Remark 3.2.10. The map ΩS 5 → BSpin is one of E 1 -spaces, so the Thom spectrum A admits the structure of an E 1 -ring with an E 1 -map A → MSpin.
Remark 3.2.11. There are multiple equivalent ways to characterize this Thom spectrum. For instance, the J-homomorphism BSpin → BGL 1 (S) sends the generator of π 4 BSpin to ν ∈ π 4 BGL 1 (S) ∼ = π 3 S. The universal property of Thom spectra in Theorem 2.1.7 shows that A is the free E 1 -ring S/ /ν with a nullhomotopy of ν. Note that A is defined integrally, and not just p-locally for some prime p.
Remark 3.2.12. There is a canonical map A → T (1) of E 1 -rings, constructed as follows. By the universal property of A, it suffices to prove that the unit S → T (1) extends along the inclusion S → Cν, i.e., that ν = 0 ∈ π 3 T (1) up to units. To see this, let us compute π 3 Cη via the exact sequence This can be identified with the final map is an isomorphism, and the first map sends η 2 → η 3 = 4ν. Therefore, π 3 Cη ∼ = Z/4{ν}. Now, since the class in H 4 (T (1); F 2 ) is detected by a nontrivial Sq 4 , the attaching map of the 4-cell in T (1) must be ±ν. Therefore, one of ±ν must be null in T (1), which implies that there must be a map Cν → T (1) (or C(−ν) → T (1)) as claimed.
The following result is [Dev19b, Proposition 2.7]; it is proved there at p = 2, but the argument clearly works for p = 3 too.
There is a map A (p) → BP. Under the induced map on BP-homology, y 2 maps to t 2 1 mod decomposables at p = 2, and to t 1 mod decomposables at p = 3. Remark 3.2.14. For instance, when p = 2, we have BP * (A) ∼ = BP * [t 2 1 + v 1 t 1 ]. One corollary (using Remark 3.2.1) is the following.
Corollary 3.2.15. As A * -comodules, we have where x 4 is a polynomial generator in degree 4.
Example 3.2.16. Let us work at p = 2 for convenience. Example 3.1.14 showed that σ 1 is the element in π 5 (Cη ∧ Cν) given by the lift of η to the 4-cell (which is attached to the bottom cell by ν) via a nullhomotopy of ην. In particular, σ 1 already lives in π 5 (Cν), and as such defines an element of S/ /ν = A (by viewing Cν as the 4-skeleton of A); note that, by construction, this element is 2-torsion. The image of σ 1 ∈ π 5 (A) under the canonical map of Remark 3.2.12 is its namesake in π 5 (T (1)). See Figure 2. Remark 3.2.17. The element σ 1 ∈ π 5 (A (2) ) defined in Example 3.2.16 in fact lifts to an element of π 5 (A), because the relation ην = 0 is true integrally, and not just 2-locally. An alternate construction of this map is the following. The Hopf map η 4 : S 5 → S 4 (which lives in the stable range) defines a map S 5 → S 4 → ΩS 5 whose composite to BSpin is null (since π 5 (BSpin) = 0). Upon Thomification of the composite S 5 → ΩS 5 → BSpin, one therefore gets a map S 5 → A whose composite with A → MSpin is null. The map S 5 → A is the element σ 1 ∈ π 5 (A).
Finally, we have: Definition 3.2.18. Let BN be the space defined by the homotopy pullback where the map f : S 13 → BO(10) detects an element of π 12 Ø(10) ∼ = Z/12. There is a fiber sequence S 9 → BO(9) → BO(10), and the image of f under the boundary map in the long exact sequence of homotopy detects 2ν ∈ π 12 (S 9 ) ∼ = Z/24. In particular, there is a fiber sequence If N is defined to be ΩBN , then there is a fiber sequence Define a map N → BString via the map of fiber sequences where the map S 9 → B 2 String detects a generator of π 8 BString. Let B denote the Thom spectrum of the induced bundle over N .
Remark 3.2.19. The map N → BString is in fact one of E 1 -spaces, so B admits the structure of an E 1 -ring. To prove this, it suffices to show that there is a map BN → B 2 String. Recall that BString = τ ≥8 Ω ∞ KO, so the desired map is the same as a class in KO 1 (BN ). Using the Serre spectral sequence for the fiber sequence defining BN , one can calculate that there is a class in KO 1 (BN ) which lifts the generator of KO 1 (S 9 ) ∼ = π 8 KO ∼ = Z. 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 is reproduced here as Figure 3. As mentioned in the introduction, the spectrum B has been briefly studied under the name X in [HM02].
Remark 3.2.20. As with A, there are multiple different ways to characterize B. There is a fiber sequence ΩS 9 → N → ΩS 13 , and the map ΩS 9 → N → BString is an extension of the map S 8 → BString detecting a generator. Under the J-homomorphism BString → BGL 1 (S), this generator maps to σ ∈ π 8 BGL 1 (S) ∼ = π 7 S, so the Thom spectrum of the bundle over ΩS 9 determined by the map ΩS 9 → BString is the free E 1 -ring S/ /σ with a nullhomotopy of ν. Proposition 2.1.6 now implies that N is the Thom spectrum of a map ΩS 13 → BGL 1 (S/ /σ). While a direct definition of this map is not obvious, we note that the restriction to the bottom cell S 12 of the source detects an element ν of π 12 BGL 1 (S/ /σ) ∼ = π 11 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 the map ν : S 11 → S 8 to Cσ determined by a nullhomotopy of σν in π * S. Althoughν ∈ π 11 Cσ does not come from a class in π 11 S, its representative in the Adams spectral sequence for Cσ is the image of h 22 in the Adams spectral sequence for the sphere.
The following result is [Dev19b, Proposition 3.2]; it is proved there at p = 2, but the argument clearly works for p ≥ 3 too.
Proposition 3.2.21. The BP * -algebra BP * (B) is isomorphic to a polynomial ring BP * [b 4 , y 6 ], where |b 4 | = 8 and |y 6 | = 12. There is a map B (p) → BP. On BP *homology, the elements b 4 and y 6 map to t 4 1 and t 2 2 mod decomposables at p = 2, and y 6 maps to t 3 1 mod decomposables at p = 3. One corollary (using Remark 3.2.1) is the following.
Corollary 3.2.22. As A * -comodules, we have where x 8 and x 12 are polynomial generators in degree 8 and 12, and b 4 is an element in degree 8.
Example 3.2.23. For simplicity, let us work at p = 2. There is a canonical ring map B → T (2), and the element σ 2 ∈ π 13 T (2) lifts to B. We can be explicit about this: the 12-skeleton of B is shown in Figure 3, and σ 2 is the element of π 13 (B) existing thanks to the relation ην = 0 and the fact that the Toda bracket η, ν, σ contains 0. This also shows that σ 2 ∈ π 13 (B) is 2-torsion.
Remark 3.2.24. The element σ 2 ∈ π 13 (B (2) ) defined in Example 3.2.23 in fact lifts to an element of π 13 (B), because the relations νσ = 0, ην = 0, and 0 ∈ η, ν, σ are all true integrally, and not just 2-locally. An alternate construction of this map S 13 → B is the following. The Hopf map η 12 : S 13 → S 12 (which lives in the stable range) defines a map S 13 → S 12 → ΩS 13 . Moreover, the composite S 13 → ΩS 13 → S 9 is null, since it detects an element of π 13 (S 9 ) = 0; choosing a nullhomotopy of this composite defines a lift S 13 → N . (In fact, this comes from a map S 14 → BN .) The composite S 13 → N → BString is null (since π 13 (BString) = 0). Upon Thomification, we obtain a map S 13 → B whose composite with B → MString is null; the map S 13 → B is the element σ 2 ∈ π 13 (B).
The following theorem packages some information contained in this section.
Proof. The existence statement for T (n) is contained in Theorem 3.1.5, while the torsion statement is the content of Lemma 3.1.16. The claims for y(n) and y Z (n) now follow from Proposition 3.2.6. The existence and torsion statements for A and B are contained in Examples 3.2.16 and 3.2.23.
The elements in Theorem 3.2.25 can in fact be extended to infinite families; this is discussed in Section 5.4.

Centers of Thom spectra.
In this section, we review some of the theory of E k -centers and state Conjecture E. We begin with the following important result, and refer to [Fra13] and [Lur16, Section 5.5.4] for proofs. . 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 modules over the factorization homology U (A) = S k−1 ×R A (known as the enveloping algebra of A), which is an E 1 -algebra in C.
, where A is regarded as a left module over its enveloping algebra via Theorem 3.3.1.
Remark 3.3.3. We are using slightly different terminology than the one used in [Lur16, Section 5.3]: our E k+1 -center is his E k -center. In other words, Lurie's terminology expresses the structure 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].  1.1]). The E k+1center Z(A) of an E k -algebra A in a symmetric monoidal presentable ∞-category C exists, and satisfies the following properties: The E k -algebra Z(A) of C defined via this universal property in fact admits the structure of an E k+1 -algebra in C. (c) There is a fiber sequence 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 presentable symmetric monoidal ∞-category C with all limits is defined inductively as the E k+m -center of the E k+m−1 -center Z k+m−1 (A). In other words, it is the universal E k+m -algebra of C which fits into a commutative diagram in Alg E k+m−1 (C).
Proposition 3.3.4 gives: Corollary 3.3.6. Let m ≥ 1. The E k+m−1 -algebra Z k+m (A) associated to an E kalgebra 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 ≥ 0 be an integer. Let R denote X(p n+1 − 1) (p) , A (in which case n = 1), or B (in which case n = 2). Then the element σ n ∈ π |σn| R lifts to the Remark 3.3.7. If R is A or B, then Z 3 (R) is the E 3 -center of the E 2 -center of R. This is a rather unwieldy object, so it would be quite useful to show that the E 1 -structure on A or B admits an extension to an E 2 -structure; we do not know if such extensions exist. Since neither ΩS 5 nor N admit the structure of a double loop space, such an E 2 -structure would not arise from their structure as Thom spectra.
In any case, if such extensions do exist, then Z 3 (R) in Conjecture E should be interpreted as the E 3 -center of the E 2 -ring R. However, we showed in [Dev19a, Theorem 4.2] that (tmf ∧ A)[x 2 ] admits an E 2 -algebra structure, where |x 2 | = 2.
Remark 3.3.8. In the introduction, we stated Conjecture 1.1.4, which instead asked about whether v n ∈ π |vn| X(p n ) lifts to π * Z 3 (X(p n )). It is natural to ask about the connection between Conjecture E and Conjecture 1.1.4. Proposition 3.1.17 implies that if Z 3 (X(p n )) admitted an X(p n − 1)-orientation factoring the canonical X(p n − 1)-orientation X(p n − 1) → X(p n ), and σ n−1 ∈ π |vn|−1 X(p n − 1) was killed by the map X(p n − 1) → Z 3 (X(p n )), then Conjecture E implies Conjecture 1.1.4. 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 3 -center of X(n) (which, recall, is the Thom spectrum of a bundle over Ω 2 BSU(n)) is Hom SU(n)+ (S, X(n)) ≃ X(n) hSU(n) , where SU(n) acts on X(n) by a Thomification of the conjugation action on ΩSU(n). . Setting G = SL n gives a description of the conjugation action of SU(n) on ΩSU(n). In light of its connections to geometric representation theory, we believe that there may be an algebro-geometric approach to proving that χ n is SU(n)-trivial in X(n) and in ΩSU(n).
Example 3.3.11. The element χ 2 ∈ π 3 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 all primes p ≥ 0. It therefore suffices to show that χ 2 is central after p-localizing for all p. First, note that χ 2 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 2 is detected in π 3 of a sphere living in dimension 3, then it could not be torsion, so it must be detected in π 3 of a sphere living in dimension 3 − k for some 0 ≤ k ≤ 2. If k = 1 or 2, then π 3 (S 3−k ) is either π 1 (S 0 ) or π 2 (S 0 ), but both of these groups vanish for p > 2. Therefore, χ 2 must be detected in π 3 of the sphere in dimension 0, i.e., in π 3 X(1). This group vanishes for p > 3, and when p = 3, it is isomorphic to Z/3 (generated by α 1 ). Since X(1) = S 0 is an E ∞ -ring, we conclude that χ 2 is central in X(2) (p) for all p > 2. At p = 2, we know the cell structure of X(2) in the bottom few dimensions (see Example 3.1.14; note that σ 1 is not χ 2 ). In dimensions ≤ 3, it is equivalent to Cη, so π 3 X(2) ∼ = π 3 Cη. However, it is easy to see that the canonical map π 3 S ≃ Z/8{ν} → π 3 Cη is surjective and exhibits an isomorphism π 3 Cη ∼ = Z/4{ν}. Therefore, χ 2 is in the image of the unit S → X(2), and is therefore vacuously central. We conclude from the above discussion that χ 2 is indeed central in X(2).
4. Review of some unstable homotopy theory 4.1. 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 > 2 and k ≥ 1 a map φ n : Ω 2 S 2n+1 → S 2n−1 (the integer k is assumed implicit) such that the composite of φ n with the double suspension E 2 : S 2n−1 → Ω 2 S 2n+1 is homotopic to the p k -th power map. By induction on n, they concluded via a result of Selick's (see [Sel77]) that p n kills the p-primary component of the homotopy of S 2n+1 . Such maps will be important in the rest of this article, so we will isolate their desired properties in the definition of a charming map, inspired by [ST19]. (Our choice of terminology is non-standard, and admittedly horrible, but it does not seem like the literature has chosen any naming convention for the sort of maps we desire.) if the composite of f with the double suspension E 2 is the degree p map, the fiber of f admits the structure of a Q 1 -space, and if there is a space BK which sits in a fiber sequence such that the boundary map Ω 2 S 2np+1 → S 2np−1 is homotopic to f .  Anick proved (see [Ani93,GT10]) that the fiber of f admits a delooping, i.e., there is a space T 2np+1 (p) (now known as an Anick space) which sits in a fiber sequence S 2np−1 → T 2np+1 (p) → ΩS 2np+1 . It follows that f is a charming map.
Remark 4.1.4. We claim that T 2p+1 (p) = ΩS 3 3 , where S 3 3 is the 3-connected cover of S 3 . To prove this, we will construct a p-local fiber sequence This fiber sequence was originally constructed by Toda in [Tod62]. To construct this fiber sequence, we first note that there is a p-local fiber sequence where the first map is the factorization of α 1 : S 2p−1 → ΩS 3 through the 2(p − 1)skeleton of ΩS 3 , and the second map is the composite J p−1 (S 2 ) → ΩS 3 → CP ∞ . This fiber sequence is simply an odd-primary version of the Hopf fibration S 3 → S 2 → CP ∞ ; the identification of the fiber of the map J p−1 (S 2 ) → CP ∞ is a simple exercise with the Serre spectral sequence. Next, we have the EHP sequence Since ΩS 3 3 is the fiber of the map ΩS 3 → CP ∞ , the desired fiber sequence is obtained by taking vertical fibers in the following map of fiber sequences: Example 4.1.5. Let W n denote the fiber of the double suspension S 2n−1 → Ω 2 S 2n+1 . Gray proved in [Gra89b,Gra88] that W n admits a delooping BW n , and that after p-localization, there is a fiber sequence 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 an Hspace structure, and is H-equivalent to BW n .
Remark 4.1.7. It has been conjectured by Cohen-Moore-Neisendorfer and Gray in the papers cited above that there is an equivalence BW n ≃ ΩT 2np+1 (p), and that ΩT 2np+1 (p) retracts off of Ω 2 P 2np+1 (p) as an H-space, where P k (p) is the mod p Moore space S k−1 ∪ p e k with top cell in dimension k. For our purposes, we shall require something slightly stronger: namely, the retraction should be one of Q 1 -spaces. The first part of this conjecture would follow from Proposition 4.1.6 if the Cohen-Moore-Neisendorfer map were a Gray map. In [Ame20], it is shown that the existence of p-primary elements of Kervaire invariant one would imply equivalences of the form BW p n−1 ≃ ΩT 2p n +1 (p).
Motivated by Remark 4.1.7 and Proposition 4.1.6, we state the following conjecture; it is slightly weaker than the conjecture mentioned in Remark 4.1.7, and is an amalgamation of slight modifications of conjectures of Cohen, Moore, Neisendorfer, Gray, and Mahowald in unstable homotopy theory, as well as an analogue of Proposition 4.1.6. (For instance, we strengthen having an H-space retraction to having a Q 1 -space retraction).
Conjecture D. The following statements are true: (a) The homotopy fiber of any charming map is equivalent as a loop space to the loop space on an Anick space. (b) There exists a p-local charming map f : Ω 2 S 2p n +1 → S 2p n −1 whose homotopy fiber admits a Q 1 -space retraction off of Ω 2 P 2p n +1 (p). There are also integrally defined maps Ω 2 S 9 → S 7 and Ω 2 S 17 → S 15 whose composite with the double suspension on S 7 and S 15 respectively is the degree 2 map, whose homotopy fibers K 2 and K 3 (respectively) admit deloopings, and which admits a Q 1 -space retraction off Ω 2 P 9 (2) and Ω 2 P 17 (2) (respectively).

Fibers of charming maps.
We shall need the following proposition.
Proof. This is an easy consequence of the Serre spectral sequence coupled with the well-known coalgebra isomorphisms where these classes are generated by the one in dimension 2n−1 via the single Dyer-Lashof operation (coming already from the cup-1 operad; see Remark 2.2.8).
with β(a 2np ) = b 2np−1 , where β is the Bockstein homomorphism. An argument with the bar spectral sequence recovers the result of Proposition 4.2.1 in this particular case.
Remark 4.2.3. Suppose that X is a space which sits in a fiber sequence such that the boundary map Ω 2 S 2np+1 → S 2np−1 has degree p j on the bottom cell of the source. The Serre spectral sequence then only has a differential on the E 2np−1 -page, and: We conclude this section by investigating Thom spectra of bundles defined over fibers of charming maps. Let R be a p-local E 1 -ring, and let µ : K → BGL 1 (R) denote a map from the fiber K of a charming map f : Ω 2 S 2np+1 → S 2np−1 . There is a fiber sequence ΩS 2np−1 → K → Ω 2 S 2np+1 of loop spaces, so we obtain a map ΩS 2np−1 → BGL 1 (R). Such a map gives an element α ∈ π 2np−3 R via the effect on the bottom cell S 2np−2 .
Theorem 2.1.7 implies that the Thom spectrum of the map ΩS 2np−1 → BGL 1 (R) should be thought of as the E 1 -quotient R/ /α, although this may not make sense if R is not at least E 2 . However, in many cases (such as the ones we are considering here), the Thom R-module R/ /α is in fact an E 1 -ring such that the map R → R/ /α is an E 1 -map. By Proposition 2.1.6, there is an induced map φ : Ω 2 S 2np+1 → BGL 1 (R/ /α) whose Thom spectrum is equivalent as an E 1 -ring to K µ . We would like to determine the element 2 of π * R/ /α detected by the restriction to the bottom cell S 2np−1 of the source of φ. First, we note: Proof. Since f is a charming map, the composite S 2np−1 → Ω 2 S 2np−1 f − → S 2np−1 is the degree p map. Therefore, the element pα ∈ π 2np−3 R is detected by the composite But there is a fiber sequence Ω 2 S 2np−1 f − → S 2np−1 → BK by the definition of a charming map, so the composite detecting pα is null, as desired.
There is now a square and the following result is a consequence of the lemma and the definition of Toda brackets: Lemma 4.2.5. The element in π 2np−2 (R/ /α) detected by the vertical map S 2np−1 → BGL 1 (R/ /α) lives in the Toda bracket p, α, 1 R/ /α .
The upshot of this discussion is the following: Proposition 4.2.6. Let R be a p-local E 1 -ring, and let µ : K → BGL 1 (R) denote a map from the fiber K of a charming map f : Ω 2 S 2np+1 → S 2np−1 , providing an element α ∈ π 2np−3 R. Assume that the Thom spectrum R/ /α of the map ΩS 2np−1 → BGL 1 (R) is an E 1 -R-algebra. Then there is an element v ∈ p, α, 1 R/ /α such that K µ is equivalent to the Thom spectrum of the map Remark 4.2.7. Let R be an E 1 -ring, and let α ∈ π d R. Then α defines a map S d+1 → BGL 1 (R), and it is natural to ask when α extends along S d+1 → ΩS 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 2 -ring, but not necessarily so if R is only an E 1 -ring. Recall that there is a cofiber sequence 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 1 (R) extends along the map S d+1 → J k (S d+1 ) if and only if there are compatible nullhomotopies of the n-fold iterated Whitehead products [α, [· · · , [α, α]], · · · ] ∈ π * BGL 1 (R) for n ≤ k. These amount to properties of Toda brackets in the homotopy of R. We note, for instance, that the Whitehead bracket [α, α] ∈ π 2d+1 BGL 1 (R) ∼ = π 2d R is the element 2α 2 ; therefore, the map S d+1 → BGL 1 (R) extends to J 2 (S d+1 ) if and only if 2α 2 = 0.
Remark 4.2.8. Let R be a p-local E 2 -ring, and let α ∈ π d (R) with d even. Then α defines an element α ∈ π d+2 B 2 GL 1 (R). The p-fold iterated Whitehead product [α, · · · , α] ∈ π p(d+2)−(p−1) B 2 GL 1 (R) ∼ = π pd+(p−1) R is given by p!Q 1 (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 1 (R). The p-fold iterated Whitehead product [α, · · · , α] ∈ π p(d+i)−(p−1) B i GL 1 (R) ∼ = π pd+(i−1)(p−1) R is given by p!Q i−1 (α) modulo decomposables. We will describe this in detail in forthcoming work: the basic idea is to reduce to the universal example of an E n -ring, and relate Whitehead products on π * (S n ) to the E d -Browder bracket on Ω d S n + (where d ≥ n). Recall the isomorphism π j S n ∼ = π j−d Ω d S n . If α ∈ π i S n and β ∈ π j S n , then we will show in future work that the stabilization of the Whitehead product [α, β] ∈ π i+j−1 S n ∼ = π i+j−d Ω d S n is closely related to the E d -Browder bracket [α, β] E d .

Chromatic Thom spectra
5.1. Statement of the theorem. To state the main theorem of this section, we set some notation. Fix an integer n ≥ 1, and work in the p-complete stable category. For each Thom spectrum R of height n−1 in Table 1, let σ n−1 : S |σn−1| → BGL 1 (R) denote a map detecting σ n−1 ∈ π |σn−1| (R) (which exists by Theorem 3.2.25). Let K n denote the fiber of a p-local charming map Ω 2 S 2p n +1 → S 2p n −1 satisfying the hypotheses of Conjecture D, and let K 2 (resp. K 3 ) denote the fiber of an integrally defined charming map Ω 2 S 9 → S 7 (resp. Ω 2 S 17 → S 15 ) satisfying the hypotheses of Conjecture D. Then: Theorem A. Let R be a height n − 1 spectrum as in the second line of Table 1. Then Conjectures D and E imply that there is a map K n → BGL 1 (R) such that the mod p homology of the Thom spectrum K µ n 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 µ n 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 Thom spectrum K µ n is equivalent to Θ(B) = tmf upon 2-completion.
We emphasize again that naïvely making sense of Theorem A relies on knowing that T (n) admits the structure of an E 1 -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 is even independent of Quillen's identification of π * MU with the Lazard ring, provided one regards the 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 future work; 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 simply Corollary 2.2.2.
Remark 5.1.3. Note that Table 2 implies that the homology of each of the Thom spectra in Table 1 are given by the Q 0 -Margolis homology of their associated designer 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. Remark 5.1.5. We can attempt to apply Theorem A for R = A in conjunction with Proposition 4.2.6. Theorem A states that Conjecture D and Conjecture E imply that there is a map K 2 → BGL 1 (A) whose Thom spectrum is equivalent to bo. There is a fiber sequence ΩS 7 → K 2 → Ω 2 S 9 , so we obtain a map µ : ΩS 7 → K 2 → BGL 1 (A). The proof of Theorem A shows that the bottom cell S 6 of the source detects σ 1 ∈ π 5 (A). A slight variation of the argument used to establish Proposition 4.2.6 supplies a map Ω 2 S 9 → B Aut((ΩS 7 ) µ ) whose Thom spectrum is bo. The spectrum (ΩS 7 ) µ has mod 2 homology F 2 [ζ 4 1 , ζ 2 2 ]. However, unlike A, it does not naturally arise an E 1 -Thom spectrum over the sphere spectrum; this makes it unamenable to study via techniques of unstable homotopy.
More precisely, (ΩS 7 ) µ is not the Thom spectrum of an E 1 -map X → BGL 1 (S) from a loop space X which sits in a fiber sequence ΩS 5 → X → ΩS 7 of loop spaces. Indeed, BX would be a S 5 -bundle over S 7 , which by [Mah87,Lemma 4] implies that X is then equivalent as a loop space to ΩS 5 × ΩS 7 . The resulting E 1 -map ΩS 7 → BGL 1 (S) is specified by an element of π 5 (S) ∼ = 0, so (ΩS 7 ) µ must then be equivalent as an E 1 -ring to A ∧ Σ ∞ + ΩS 7 . In particular, σ 1 ∈ π 5 (A) would map nontrivially to (ΩS 7 ) µ , which is a contradiction.
The proof of Theorem A will also show: Corollary 5.1.6. Let R be a height n − 1 spectrum as in the second line of Table  1, and assume Conjecture F if R = B. Let M be an E 3 -R-algebra. Conjecture D and Conjecture E imply that if:

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 down into multiple steps. The result for y(n) and y Z (n) follow from the result for T (n) by Proposition 3.2.6, 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 2 and K 3 denote the integrally defined spaces from Conjecture D. By Remarks 3.2.17 and 3.2.24, the elements σ 1 ∈ π 5 (A) and σ 2 ∈ π 13 (B) are defined integrally. We will write σ n−1 to generically denote this element, and will write it as living in degree |σ n−1 |. 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 hope that this is worth circumventing the task of having to read through essentially the same proofs for these slightly different cases.
Unfortunately, this is not true; but this is where Conjecture E comes in: it says that the element σ n−1 ∈ π |σn−1| R lifts to the E 3 -center Z 3 (R), where it has the same torsion order as in R. (Here, we are abusively writing Z 3 (T (n − 1)) to denote the E 3 -center of X(p n − 1) (p) .) The lifting of σ n−1 to π |σn−1| Z 3 (R) provided by Conjecture E gives a factorization of the map from (5.1) as Since Z 3 (R) is an E 3 -ring, BGL 1 (Z 3 (R)) admits the structure of an E 2 -space. In particular, the map P |σn−1|+2 (p) → BGL 1 (Z 3 (R)) factors through Ω 2 P |σn−1|+4 (p), as desired. We let µ denote the resulting composite Step 2. Theorem A asserts that there is an identification between the Thom spectrum of the induced map µ : K n → BGL 1 (R) and the associated designer chromatic spectrum Θ(R) via Table 1. We shall identify the Steenrod comodule structure on the mod p homology of K µ n , and show that it agrees with the mod p homology of Θ(R).

Designer chromatic spectrum
Mod p homology  [AR05,Proposition 5.3] for a proof of the statement for H * (BP n − 1 ; F p ); this implies the calculations of H * (k(n−1); F p ) and H * (k Z (n − 1); F p ). See [AR05, Proposition 6.1] for a proof of the statements for H * (bo; F 2 ) and H * (tmf; F 2 ), and [Rez07, Theorem 21.5] for H * (tmf; F p ) for any p. For odd p, bo (p) is a sum of shifts of BP 1 , which implies the statement about H * (bo; F p ).
In Table 3, we have recorded the mod p homology of the designer chromatic spectra in Table 1 (see [LN14,Theorem 4.3] for BP n − 1 ). It follows from Proposition 4.2.1 that there is an isomorphism Combining this isomorphism with Theorem 3.1.5, Proposition 3.2.4, Proposition 3.2.15, and Proposition 3.2.22, we find that there is an abstract equivalence between the mod p homology of K µ n and the mod p homology of Θ(R).
Step 3. By Step 2, the mod p homology of the Thom spectrum K µ n is isomorphic to the mod p homology of the associated designer chromatic spectrum Θ(R) as a Steenrod comodule. The main result of [AL17] and [AP76, Theorem 1.1] now imply that unless R = B, the Thom spectrum K µ n is equivalent to Θ(R) upon p-completion for every prime p. Finally, if Conjecture F is true, then the same conclusion can be drawn for B: the Thom spectrum K µ n is equivalent to Θ(B) = tmf upon p-completion for every prime p.
This concludes the proof of Theorem A.
5.3. Remark on the proof. Before proceeding, we note the following consequence of the proof of Theorem A.
5.4. Infinite families and the nilpotence theorem. We now briefly discuss the relationship between Theorem A and the nilpotence theorem. We begin by describing a special case of this connection. Recall from Remark 2.2.3 that Theorem 2.2.1 implies that if R is an E 2 -ring spectrum, 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 A when n = 1) implies that if R is a p-local E 3 -ring spectrum, and x ∈ π * R is a class with trivial HZ p -Hurewicz image such that: • α 1 x = 0 in π * R; and • the Toda bracket p, α 1 , x contains zero; then x is nilpotent.
Proof. We claim that the composite is null. Remark 4.1.8 implies that Conjecture D for n = 1 reduces us to showing that the composite is null. Since this composite is one of double loop spaces, it further suffices to show that the composite is null. The bottom cell S 2p−2 of P 2p−1 (p) maps trivially to BGL 1 (R[1/x]), because the bottom cell detects α 1 (by Remark 4.1.8), and α 1 is nullhomotopic in R[1/x]. Therefore, the map (5.3) factors through the top cell S 2p−1 of P 2p−1 (p). The resulting map S 2p−1 → BGL 1 (S (p) ) → BGL 1 (R[1/x]) detects an element of the Toda bracket p, α 1 , x , but this contains zero by hypothesis, so is nullhomotopic.
Since the map (5.2) is null, Corollary 2.2.2 and Theorem 2.1.7 implies that there is a ring map HZ 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: The bottom map, however, is null, because x has zero HZ p -Hurewicz image. Therefore, the element x ∈ π * R[1/x] is null, and hence R[1/x] is contractible.
Remark 5.4.2. One can prove by a different argument that Proposition 5.4.1 is true without the 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 following proposition describes the construction of some infinite families.
Proposition 5.4.3. Let R be a height n − 1 spectrum as in the second line of Table 2, and assume Conjecture E if R = A or B. Then there is an infinite family σ n−1,p k ∈ π p k |vn|−1 (R). Conjecture E implies that σ n−1,p k lifts to π p k |vn|−1 (Z 3 (R)), where Z 3 (R) abusively denotes the E 3 -center of X(p n − 1) if R = T (n − 1).
Proof. We construct this family by induction on k. The element σ n−1,1 is just σ n−1 , so assume that we have defined σ n−1,p k . The element σ n−1,p k ∈ π p k |vn|−1 R defines a map σ n−1,p k : S p k |vn| → BGL 1 (R). When R = T (n − 1), Lemma 3.1.12 (and the inductive hypothesis) implies that the map defined by σ n factors through the map BGL 1 (X(p n − 1)) → BGL 1 (T (n − 1)). When R = A or B, Conjecture E (and the inductive hypothesis) implies that the map defined by σ n factors through the map BGL 1 (Z 3 (R)) → BGL 1 (R). This implies that for all R as in the second line of Table 2, the map σ n−1,p k : S p k |vn| → BGL 1 (R) factors through an E 1 -space, which we shall just denote by Z R for the purpose of this proof. If we assume Conjecture E, then we may take Z R = BGL 1 (Z 3 (R)).
Remark 5.4.4. This infinite family is detected in the 1-line of the ANSS for R by δ(v k n ), where δ is the boundary map induced by the map Σ −1 R/p → R. This is a consequence of the geometric boundary theorem (see [Rav86,Theorem 2.3.4]) applied to the cofiber sequence R p − → R → R/p.
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 − 1)module skeleta (denoted G k in [DHS88]) of T (n) are nilpotent. Let us assume that p > 2 for simplicity. Then these self-maps are given by multiplication by the p-fold Toda bracket b n,k = σ n−1,p k , · · · , σ n−1,p k at an odd prime p; this lives in degree p|σ n−1,p k | + p − 2 = 2p k (p n − 1) − 2. (When p = 2, the desired element σ n−1,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 Conjecture D and Conjecture E for the remainder of this discussion. The motivation for this approach stems from the observation that if R is any E 3 -F 2 -algebra and x ∈ π * (R), then there is a relation Q 1 (x) 2 = Q 2 (x 2 ) (at odd primes, one has a relation involving the p-fold Toda bracket Q 1 (x), · · · , Q 1 (x) ). In our setting, Proposition 5.4.3 implies that the elements σ n−1,k lift to π * Z 3 (X(p n − 1)). At p = 2, one can prove (in the same way that the Cartan relation Q 1 (x) 2 = Q 2 (x 2 ) is proven) that the construction of this infinite family implies that σ 2 n−1,p k+1 can be described in terms of Q 2 (σ 2 n−1,p k ). At odd primes, there is a similar relation involving the p-fold Toda bracket defining b n,k . In particular, induction on k implies that the b n,k are all nilpotent in π * Z 3 (X(p n − 1)) if b n,1 is nilpotent. Note that |b n,1 | = 2p n+1 − 2p − 2.
To argue that b n,1 is nilpotent, one first observes that σ n−1 b p n,1 = 0 in π * Z 3 (X(p n − 1)); when n = 0, this follows from the statement that α 1 β p 1 = 0 in the sphere. To show that b n,1 is nilpotent, it suffices to establish that Z 3 (X(p n − 1))[1/b p n,1 ] is contractible; when n = 1, this follows from Proposition 5.4.1. We give a very brief sketch of this nilpotence for general n, by arguing as in Proposition 5.4.1, and with a generous lack of precision which will be remedied in forthcoming work.
6. Applications 6.1. Splittings of cobordism spectra. The goal of this section is to prove the following.
Theorem C. Assume that the composite Z 3 (B) → B → MString (2) is an E 3map. Then Conjectures 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 3 (B) → B → MString (2) is an E 3 -map is too strong: we believe that it can be removed using special properties of fibers of charming maps, and we will return to this in future work.
We only construct unstructured splittings; it seems unlikely that they can be refined to structured 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 3 (A) → A → MSpin (2) is an E 3 -map, then Conjecture D and Conjecture E imply that there are unital splittings of the Atiyah-Bott-Shapiro orientation MSpin (2) → bo (2) . This splitting was originally proved unconditionally (i.e., without assuming Conjecture D or Conjecture E) by Anderson-Brown-Peterson in [ABP67] via a calculation with the Adams spectral sequence. Remark 6.1.5. In [Dev19b], we proved (unconditionally) that the map π * MString → π * tmf is surjective. Our proof proceeds by showing that the map π * B → π * tmf is surjective via arguments with the Adams-Novikov spectral sequence and by exploiting the E 1 -ring structure on B to lift the powers of ∆ living in π * tmf.
The discussion preceding [MR09,Remark 7.3] (in the arXiv version of the document) implies that for a particular model of tmf 0 (3), we have: Corollary 6.1.6. Assume that the composite Z 3 (B) → B (2) → MString (2) is an E 3 -map. Then Conjectures D, E, and F imply that Σ 16 tmf 0 (3) ∧ 2 is a summand of MString ∧ 2 . We now turn to the proof of Theorem C.
Proof of Theorem C. First, note that such a splitting exists after rationalization. Indeed, it suffices to check that this is true on rational homotopy; since the orientations under considerations are E ∞ -ring maps, the induced map on homotopy is one of rings. It therefore suffices to lift the generators.
We now show that the generators of π * tmf ⊗Q ∼ = Q[c 4 , c 6 ] lift to π * MString⊗Q. Although one can argue this by explicitly constructing manifold representatives (as is done for c 4 in [Dev19b, Corollary 6.3]), it is also possible to provide a more homotopy-theoretic proof: the elements c 4 and c 6 live in dimensions 8 and 12 respectively, and the map MString → tmf is known to be an equivalence in dimensions ≤ 15 by [Hil,Theorem 2.1]. It follows that the same is true rationally, so c 4 and c 6 indeed lift to π * MString ⊗ Q, as desired.
We will now construct a splitting after p-completion where p = 2. By Corollary 5.1.6, we obtain a unital map tmf ≃ Θ(B) → MString upon p-completion which splits the orientation MString → Θ(B) because: (a) the map Z 3 (B) → B → MString is an E 3 -ring map (by assumption).
Remark 6.1.7. The proof recalled in Remark 1.1.2 of Thom's splitting of MO proceeded essentially unstably: there is an E 2 -map Ω 2 S 3 → BO of spaces over BGL 1 (S), whose Thomification yields the desired E 2 -map HF 2 → MO. This argument also works for MSO: there is an E 2 -map Ω 2 S 3 3 → BSO of spaces over BGL 1 (S), whose Thomification yields the desired E 2 -map HZ → MSO. One might hope for the existence of a similar unstable map which would yield 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 the discussion is the same for MSpin.
According to Theorem A, Conjecture D and Conjecture E imply that there is a map K 3 → BGL 1 (B) whose Thom spectrum is equivalent to tmf. There is a map BN → B 2 String, whose fiber 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 1 (B) whose Thom spectrum is MString. Theorem C would follow if there was a map f : K 3 → Q of spaces over BGL 1 (B), since Thomification would produce a map tmf → MString.
Conjecture D reduces the construction of f to the construction of a map Ω 2 P 17 (2) → Q. This map would in particular imply the existence of a map P 15 (2) → Q (and would be equivalent to the existence of such a map if Q was a double loop space), which in turn stems from a 2-torsion element of π 14 (Q). The long exact sequence on homotopy runs · · · → π 14 (BString) → π 14 (Q) → π 13 (N ) → π 13 (BString) → · · · Bott periodicity states that π 13 BString ∼ = π 14 BString ∼ = 0, so we find that π 14 (Q) ∼ = π 13 (N ). The desired 2-torsion element of π 14 (Q) is precisely the element of π 13 (N ) described in Remark 3.2.24. Choosing a particular nullhomotopy of twice this 2torsion element of π 14 (Q) produces a map g : P 15 (2) → Q. To extend this map over the double suspension P 15 (2) → Ω 2 P 17 (2), it would suffice to show that there is a double loop space Q with a map Q → Q such that g factors through Q.
Unfortunately, we do not know how to prove such a result; this is the unstable analogue of Conjecture E. In fact, such an unstable statement would bypass the need for Conjecture E in Theorem A. (One runs into the same obstruction for MSpin, except with the fiber of the map S 5 → B 2 Spin.) These statements are reminiscent of the conjecture (see Section 4.1) that the fiber W n = fib(S 2n−1 → Ω 2 S 2n+1 ) of the double suspension admits the structure of a double loop 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-Shapiro orientation MSpin → KO induces an isomorphism of KO * -modules for all spectra X. In future work, we shall show that Theorem C can be used to prove the following height 2 analogue of this result: namely, Conjectures D, E, and F imply that the Ando-Hopkins-Rezk orientation MString → Tmf induces an isomorphism of Tmf * -modules for all spectra X. The K(1)-analogue of this isomorphism was obtained by Laures in [Lau04].
Our goal in this section is to revisit these Wood equivalences using the point of view stemming from Theorem A. In particular, we propose that these equivalences are suggested by the existence of certain EHP sequences; we will greatly expand on this in a forthcoming document. We find this to be a rather beautiful connection between stable and unstable homotopy theory.
The first Wood-style result was proved in Proposition 3.2.6. The next result, originally proved in [Mah79, Section 2.5] and [DM81, Theorem 3.7], is the simplest example of a Wood-style equivalence which is related to the existence of certain EHP sequences.
Proposition 6.2.1. Let S/ /η = X(2) (resp. S/ /2) denote the E 1 -quotient of S by η (resp. 2). If Y = Cη ∧ S/2 and A 1 is a spectrum whose cohomology is isomorphic to A(1) as a module over the Steenrod algebra, then there are equivalences Remark 6.2.2. Proposition 6.2.1 implies the Wood equivalence bo ∧ Cη ≃ bu. Although this implication is already true before 2-completion, we will work in the 2-complete category for convenience. Recall that Theorem A states that Conjecture D and Conjecture E imply that there is a map µ : K 2 → BGL 1 (A) whose Thom spectrum is equivalent to bo (as left A-modules). Moreover, the Thom spectrum of the composite K 2 µ − → BGL 1 (A) → BGL 1 (T (1)) is equivalent to BP 1 . Since this Thom spectrum is the base-change K µ 2 ∧ A T (1), and Proposition 6.2.1 implies that T (1) = X(2) ≃ A ∧ Cη, we find that Similarly, noting that S/ /2 = y(1), we find that Proposition 6.2.1 also proves the equivalence bo ∧ Y ≃ k(1).
Remark 6.2.3. The argument of Remark 6.2.2 in fact proves that Theorem A for A implies Theorem A for T (1), y Z (1), and y(1).
Proof of Proposition 6.2.1. For the first two equivalences, it suffices to show that A ∧ Cη ≃ S/ /η and that S/ /η ∧ S/2 ≃ S/ /2. We will prove the first statement; the proof of the second statement is exactly the same. There is a map Cη → S/ /η given by the inclusion of the 2-skeleton. There is also an E 1 -ring map A → S/ /η given as follows. The multiplication on S/ /η defines a unital map Cη ∧ Cη → S/ /η. But since the Toda bracket η, 2, η 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 1 -ring map A → S/ /η.
For the final equivalence, it suffices to construct a map A 1 → y(1)/v 1 for which the induced map A ∧ A 1 → y(1)/v 1 gives an isomorphism on mod 2 homology. Since A 1 may be obtained as the cofiber of a v 1 -self map Σ 2 Y → Y , it suffices to observe that the the following diagram commutes; our desired map is the induced map on vertical cofibers: Remark 6.2.4. There are EHP sequences Recall that S/2, Cη, S/ /2, S/ /η = X(2), and A are Thom spectra over S 1 , S 2 , ΩS 2 , ΩS 3 , and ΩS 5 respectively. Proposition 2.1.6 therefore implies that there are maps f : ΩS 3 → B Aut(S/2) and g : ΩS 5 → B Aut(Cη) whose Thom spectra are equivalent to S/ /2 and S/ /η, respectively. The maps f and g define local systems of spectra over ΩS 3 and ΩS 5 whose fibers are equivalent to S/2 and Cη (respectively), Corollary 6.2.11. If the telescope conjecture is true for F (and hence any type 1 or 2 spectrum) or R, then it is true for R ′ .
Proof. Since L n -and L f n -localizations are smashing, we find that if the telescope conjecture is true for F or R, then Propositions 6.2.1 and 6.2.7 yield equivalences Finally, we prove Proposition 6.2.7.
Proof of Proposition 6.2.7. We first construct maps B → T (2) and DA 1 → T (2). The top cell of DA 1 is in dimension 12, and the map T (2) → BP is an equivalence in dimensions ≤ 12. It follows that constructing a map DA 1 → T (2) is equivalent to constructing a map DA 1 → BP. However, both BP and DA 1 are concentrated in even degrees, so the Atiyah-Hirzebruch spectral sequence collapses, and we find that BP * (DA 1 ) ∼ = H * (DA 1 ; BP * ). The generator in bidegree (0, 0) produces a map DA 1 → T (2); its effect on homology is the additive inclusion F 2 [ζ 2 1 , ζ 2 2 ]/(ζ 8 1 , ζ 4 2 ) → F 2 [ζ 2 1 , ζ 2 2 ]. The map B → T (2) may be defined via the universal property of Thom spectra from Section 2.1 and Remark 3.2.20. Its effect on homology is the inclusion . We obtain a map B ∧ DA 1 → T (2) via the multiplication on T (2), and this induces an isomorphism in 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: As an aside, we note that the element σ 1 is intimately related to the element discussed in Example 3.1.14; namely, it is given by the self-map of A 1 ∧ Cν given by smashing A 1 with the following diagram: O O Using these cofiber sequences and Proposition 3.2.6, one obtains a map Z → y(2), which induces the additive inclusion F 2 [ζ 1 , ζ 2 ]/(ζ 8 1 , ζ 4 2 ) → F 2 [ζ 1 , ζ 2 ] on mod 2 homology. The multiplication on 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 2 → y(2)/v 2 for which the induced map B ∧ A 2 → y(2)/v 2 gives an isomorphism on mod 2 homology. Since A 2 may be obtained as the cofiber of a v 2 -self map Σ 6 Z → Z, it suffices to observe that the the following diagram commutes; our desired map is the induced map on vertical cofibers: Arguing exactly as in the proof of Proposition 6.2.7 shows the following result at the prime 3: Proposition 6.2.12. Let X 3 denote the 8-skeleton of T (1) = S/ /α 1 . There are 3-complete equivalences In forthcoming work, we will discuss the relation between Proposition 6.2.7 and EHP sequences, along the lines of Remark 6.2.4.

C 2 -equivariant analogue of Corollary B
Our goal in this section is to study a C 2 -equivariant analogue of Corollary B at height 1. The odd primary analogue of this result is deferred to the future; it is considerably more subtle.
7.1. C 2 -equivariant analogues of Ravenel's spectra. In this section, we construct the C 2 -equivariant analogue of T (n) for all n. We 2-localize everywhere until mentioned otherwise. There is a C 2 -action on ΩSU(n) given by complex conjugation, and the resulting C 2 -space is denoted ΩSU(n) R . Real Bott periodicity gives a C 2 -equivariant map ΩSU(n) R → BU R whose Thom spectrum is the (genuine) C 2 -spectrum X(n) R . This admits the structure of an E ρ -ring, since it is the Thom spectrum of an E ρ -map Ω ρ B σ SU(n) R → Ω ρ B ρ BU R ≃ Ω ρ BSU R . 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 ≤ 2 n+1 − 1. Again, this summand admits the structure of an E 1 -ring.
Construction 7.1.1. There is an equivariant fiber sequence where ρ is the regular representation of C 2 ; the equivariant analogue of Proposition 2.1.6 then shows that there is a map ΩS nρ+1 → BGL 1 (X(n) R ) (detecting an element χ n ∈ π nρ−1 X(n) R ) whose Thom spectrum is X(n+1) R . Here, BGL 1 (X(n) R ) is the delooping of the E ρ -space GL 1 (X(n) R ), and the C 2 -equivariant notion of Thom spectrum is taken in the sense of [HHK + 20, Theorem 3.2]. (The constructions from loc. cit. can be verified to go through for equivariant maps to BGL 1 (X(n) R ); for example, when n = ∞, the idea of taking Thom spectra for an equivariant map to BGL 1 (MU R ) was already used in [HS20, Section 3].) If σ n denotes the image of the element χ 2 n+1 ρ−1 in π (2 n+1 −1)ρ−1 T (n) R , then we have a C 2 -equivariant analogue of Lemma 3.1.12: Lemma 7.1.2. The Thom spectrum of the map ΩS (2 n+1 −1)ρ+1 → BGL 1 (X(2 n+1 − 1) R ) detecting σ n is a direct sum of shifts of T (n + 1) R .
Example 7.1.3. For instance, T (1) R = X(2) R is the Thom spectrum of the map ΩS ρ+1 → BU R ; upon composing with the equivariant J-homomorphism BU R → BGL 1 (S), this detects the element η ∈ π σ S, and the extension of the map S ρ → BGL 1 (S) to ΩS ρ+1 uses the E 1 -structure on BGL 1 (S). The case of X(2) R exhibits a curious property: S ρ+1 is the loop space Ω σ HP ∞ R , and there are equivalences (see [HW20, Proposition 3.4 and Proposition 3.6]) . However, ΩHP ∞ R ≃ S ρ+σ , so ΩS ρ+1 = Ω σ S ρ+σ . The map Ω σ S ρ+σ → BGL 1 (S) still detects the element η ∈ π σ S on the bottom cell, but the extension of the map S ρ → BGL 1 (S) to Ω σ S ρ+σ is now defined via the E σ -structure on BGL 1 (S). The upshot of this discussion is that X(2) R is not only the free E 1 -ring with a nullhomotopy of η, but also the free E σ -algebra with a nullhomotopy of η.
In other words, there is a fiber sequence S σ → S ρ+σ η − → S ρ . On geometric fixed points, this produces the fiber sequence S 0 = C 2 → S 1 → S 1 , which forces the map Φ C2 η to have degree 2 (or −2, depending on the choice of orientation).
As an aside, we mention that there is a C 2 -equivariant lift of the spectrum A: Definition 7.1.6. Let A C2 denote the Thom spectrum of the map ΩS 2ρ+1 → BGL 1 (S) defined by the extension of the map S 2ρ → BGL 1 (S) which detects the equivariant Hopf map ν ∈ π 2ρ−1 S.
We now prove the proposition used above.
In fact, it is easy to prove the following analogue of Proposition 6.2.1: Proof. There are maps A C2 → T (1) R and C η → T (1) R , which define a map A C2 ∧ C 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/2 → S/ /2. This was also proved in the course of Proposition 6.2.1.
Remark 7.1.11. As in Remark 6.2.2, one might hope that this implies the C 2equivariant Wood equivalence bo C2 ∧ C 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 Wood equivalence of Proposition 7.1.10 stems from the EHP sequence S ρ → ΩS ρ+1 → ΩS 2ρ+1 . To prove the existence of such a fiber sequence, we use [DH21,Construction 4.26] to get the Hopf map h : ΩS ρ+1 → ΩS 2ρ+1 , as well as a nullhomotopy of the composite S ρ → ΩS ρ+1 → ΩS 2ρ+1 . In particular, if F = fib(h), there is an equivariant map S ρ → F . We claim that this map is an equivalence: it suffices to prove that S ρ → F is an equivalence on underlying and on geometric fixed points, since these functors preserve homotopy limits and colimits, and these functors are jointly conservative. The desired equivalence on underlying spaces follows from the classical EHP sequence S 2 → ΩS 3 → ΩS 5 , and the equivalence on geometric fixed points follows from the splitting ΩS 2 ≃ S 1 ×ΩS 3 . 7.2. The C 2 -equivariant analogue of Corollary B at n = 1. Recall (see [HK01]) that there are indecomposable classes v n ∈ π (2 n −1)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 2ρ+1 → BGL 1 (T (1) R ) detecting an indecomposable in π ρ T (1) R on the bottom cell, whose Thom spectrum is HZ.
Note that, as with Corollary B at n = 1, this result is unconditional. The argument is exactly as in the proof of Corollary B at n = 1, with practically no modifications. We need the following analogue of Theorem 2.2.1, originally proved in [BW18,HW20].
We can now prove Theorem 7.2.1.
To identify the fibers, note that there is the Hopf fiber sequence S ρ+σ η − → S ρ → CP ∞ R . The fiber of the middle vertical map is ΩS ρ+1 ρ+1 via the definition of S ρ+1 ρ+1 as the homotopy fiber of the map S ρ+1 → BCP ∞ R . It remains to show that the map Ω ρ S 2ρ+1 → BGL 1 (T (1) R ) detects an indecomposable element of π ρ T (1) R . Indecomposability in π ρ T (1) R ∼ = π ρ BP R is the same as not being divisible by 2, so we just need to show that the dotted map in the following diagram does not exist: x x q q q q q q BGL 1 (T (1) R ) If this factorization existed, there would be an orientation HZ → T (1) R , which is absurd.
We now explain why we do not know how to prove the equivariant analogue of Corollary B at higher heights. One could propose an equivariant analogue of Conjecture D, and such a conjecture would obviously be closely tied with the existence of some equivariant analogue of the work of Cohen-Moore-Neisendorfer. We do not know if any such result exists, but it would certainly be extremely interesting.
Suppose that one wanted to prove a result like Corollary B, stating that the equivariant analogues of Conjecture D and Conjecture E imply that there is a map Ω ρ S 2 n ρ+1 → BGL 1 (T (n) R ) detecting an indecomposable in π (2 n −1)ρ T (n) R on the bottom cell, whose Thom spectrum is BP n − 1 R . One could then try to run the same proof as in the nonequivariant case by constructing a map from the fiber of a charming map Ω ρ S 2 n ρ+1 → S (2 n −1)ρ+1 to BGL 1 (T (n − 1) 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 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 equivariant setting. In future work, we shall describe a related project connecting the T (n) spectra to the Andrews-Gheorghe-Miller w nperiodicity in C-motivic homotopy theory (see [AM17,Ghe17,Kra17]).
However, since there is a map Ω λ S λ+1 λ + 1 → BGL 1 (S) as in Proposition 7.2.2, there may nevertheless be a way to construct a suitable map from the fiber of a charming map Ω ρ S 2 n ρ+1 → S (2 n −1)ρ+1 to BGL 1 (T (n− 1) R ). Such a construction would presumably provide a more elegant construction of the nonequivariant map used in the proof of Theorem A.

Future directions
In this section, we suggest some directions for future investigation. This is certainly not an exhaustive list; there are numerous questions we do not know how to address that are spattered all over this document, but we have tried to condense some of them into the list below. We have tried to order the questions in order of our interest in them. We have partial progress on many of these questions.
(a) Some obvious avenues for future work are the conjectures studied in this article: Conjectures D, E, F, and 3.1.7. Can the E 3 -assumption in the statement of Theorem C be removed? (b) One of the Main Goals TM of this project is to rephrase the proof of the nilpotence theorem from [DHS88,HS98]. As mentioned in Remark 2.2.3, the Hopkins-Mahowald theorem for HF p immediately implies the nilpotence theorem for simple p-torsion classes in the homotopy of a homotopy commutative ring spectrum (see also [Hop84]). We will expand on the relation between the results of this article and the nilpotence theorem in forthcoming work; see Remark 5.4.7 for a sketch. From this point of view, Theorem A is very interesting: it connects torsion in the unstable homotopy groups of spheres (via Cohen-Moore-Neisendorfer) to nilpotence in the stable homotopy groups of spheres. We are not sure how to do so, but could the Cohen-Moore-Neisendorfer bound for the exponents of unstable homotopy groups of spheres be used to obtain bounds for the nilpotence exponent of the stable homotopy groups of spheres? (c) It is extremely interesting to contemplate the interaction between unstable homotopy theory and chromatic homotopy theory apparent in this article. Connections between unstable homotopy theory and the chromatic picture have appeared elsewhere in the literature (e.g., in [AM99, Aro98, Mah82, MT94]), but their relationship to the content of this project is not clear to me. It would be interesting to have this clarified. One naïve 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 of slope 1/|σ n | (see [Mah81a] for the case R = A). Can another proof of this vanishing line be given using general arguments involving Thom spectra? We have some results in this direction 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 the corresponding designer spectrum on the third line are surjective on homotopy. In the case of tmf, this requires some computational effort to prove, and has been completed in [Dev19b]. This behavior is rather unexpected: in general, the unit map from a structured ring to some structured quotient will not be surjective on homotopy. Is there a conceptual reason for this surjectivity? (f) In [BBB + 19], the tmf-resolution of a certain type 2 spectrum Z is studied.
Mahowald uses the Thom spectrum A to study the bo-resolution of the sphere in [Mah81a], so perhaps the spectrum B could be used to study the tmf-resolution of Z. This is work in progress. See also Corollary 6.2.11 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 A produces "relative" Brown-Gitler spectra for BP n , bo, and tmf. In future work, we will study these spectra and show how they relate to the Davis-Mahowald non-splitting of 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 hope for 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 Thom spectra 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 → HZ is surjective on homotopy. The Kahn-Priddy theorem, stating that the transfer λ : Σ ∞ RP ∞ → S is surjective on π * ≥1 , can be interpreted as stating that π * Σ ∞ RP ∞ contains those elements of π * S which are not detected by HZ. 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 along with a map λ R : P → R such that each x ∈ π * R in the kernel of the map R → Θ(R) lifts 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 ∞ RP ∞ ≃ Σ −1 Sym 2 (S)/S, the existence of such a result is very closely tied to an analogue of the Whitehead conjecture (see [Kuh82]; the Whitehead conjecture implies the Kahn-Priddy theorem). In particular, one might expect the answer to the question posed 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 spectrum construction (via Table 1) of the conjectural spectrum eo p−1 ? Such a spectrum would be the Thom spectrum of a p-complete spherical fibration over a p-local space built via p − 1 fiber sequences from the loop spaces ΩS 2k(p−1)+1 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 welldefined since T (n) is homotopy commutative) 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 1 , · · · , v n−1 . What are the moduli problems classified by A and B? We do not know if this question even makes sense at p = 2, since A and B are a priori only E 1 -rings. Nonetheless, in [Dev19a], we provide a description of tmf ∧A in terms of the Hodge filtration of the universal elliptic curve (even at p = 2); we also showed that (tmf ∧ A)[x 2 ] admits an E 2 -algebra structure, where |x 2 | = 2. (m) Theorem A shows that the Hopkins-Mahowald theorem for HZ p can be generalized to describe forms of BP n ; at least for small n, these spectra have associated algebro-geometric interpretations (see [DFHH14,Chapter 9]). What is the algebro-geometric interpretation of Theorem A?