Elsevier

Advances in Mathematics

Volume 387, 27 August 2021, 107839
Advances in Mathematics

Real topological Hochschild homology and the Segal conjecture

https://doi.org/10.1016/j.aim.2021.107839Get rights and content

Abstract

We give a new proof, independent of Lin's theorem, of the Segal conjecture for the cyclic group of order two. The key input is a calculation, as a Hopf algebroid, of the Real topological Hochschild homology of F2. This determines the E2-page of the descent spectral sequence for the map NF2F2, where NF2 is the C2-equivariant Hill–Hopkins–Ravenel norm of F2. The E2-page represents a new upper bound on the RO(C2)-graded homotopy of NF2, from which the Segal conjecture is an immediate corollary.

Introduction

The Segal conjecture, for the cyclic group C2 of order 2, is an equivalenceπs0(RP)Aˆ(C2) of the stable cohomotopy of RP with the completion of the Burnside ring of C2 at the augmentation ideal. The conjecture follows from the following stronger result of Lin [16]:

Theorem 1.1 Lin

Let γ denote the canonical line bundle over RP, and for each integer n>0 let RPn denote the Thom spectrum of nγ. Then there is an equivalence of spectraRP=holimnRPn(S1)2.

The only known proof of Lin's theorem proceeds via calculation of a certain continuous Ext groupExtˆA(H(RP;F2),F2). The calculation is elegant, and has been generalized through the development of the Singer construction [19], [1], [17]. However, the simplicity of Lin's proof is fundamentally limited by the complexity of the Steenrod algebra A. The goal of this paper is to provide a new, less computational proof of Lin's theorem. We cannot avoid calculating a completed Ext group, but the Ext we calculate is over a polynomial coalgebra F2[x] rather than the Steenrod algebra A. We trust the reader will agree that this reduces the complexity of the homological algebra.

Remark 1.2

Just as the Steenrod algebra A arises as (the dual of) the homology of F2, the polynomial coalgebra F2[x] appears as the topological Hochschild homology of F2.

To explain our methods, we must review how the Segal conjecture has been both restated and generalized via the language of C2-equivariant stable homotopy theory.

Notation 1.3

In the C2-equivariant stable homotopy category, we use the notation S=S0 to denote the unit object. This is the C2-equivariant sphere spectrum. We use Sσ to denote the 1-point compactification of the sign representation. Depending on context, we use F2 to denote either the field with 2 elements or the non-equivariant Eilenberg–Maclane spectrum HF2.

Recollection 1.4

In the C2-equivariant stable homotopy category, the morphisma:SσS0 is adjoint to the inclusion of fixed points into the sign representation σ. The Borel completion of a C2-spectrum X is the a-completionXa:=holim(X/anX/an1X/a). One says that a C2-spectrum X is Borel complete if the natural map XXa is an equivalence.

Theorem 1.5 Lin's theorem, restated

The natural map SSa is an equivalence after 2-completion.

We will explain the equivalence of the two variants of Lin's theorem in Section 5. In the above form, Lin's theorem has received a substantial generalization.

Recollection 1.6

For any ordinary spectrum X, the Hill–Hopkins–Ravenel norm NX=NeC2X is a C2-equivariant refinement of the smash product XX, with C2-action given by swapping the two copies of X [11, §B.5].

A version of the following was first proved in [14] (cf. [17, Theorem 5.13]). As we will recall in Section 5, the statement in full generality is a consequence of [20, III.1.7].

Theorem 1.7 Segal conjecture, strong form

Let X denote any bounded below spectrum. Then the natural mapNX(NX)a is an equivalence after 2-completion.

Theorem 1.5 follows from Theorem 1.7 by setting X to be the sphere spectrum. As explained in [20, III.1.7], Theorem 1.7 follows in general from the case X=F2. In other words, since NF2 is 2-complete, all statements of Lin's theorem are consequences of the following result:

Theorem 1.8

The C2-spectrum NF2 is Borel complete.

Theorem 1.8 is the form in which we will prove the Segal conjecture. It is important to note that, while Theorem 1.8 tells us that the spectra NF2 and (NF2)a coincide, it does not shed light on the homotopy type of either one. As we now explain, our main theorem provides a computable upper bound on the homotopy groups of these spectra, and in this sense our results are stronger than the Segal conjecture.

We prove the following Theorem and Corollary independently of the Segal conjecture. From here on, all Hopf algebras, comodules, and homotopy groups will be indexed over RO(C2), the virtual representation ring of C2. The functor Ext is then defined using relative injective resolutions in this category of RO(C2)-graded comodules, just as in [18, §A.1].

Theorem A

Let F2[x] be the Hopf algebra with x primitive of degree 1+σ, and let F2[a,u] be the comodule algebra where the class a is primitive in degreeσ, u is in degree 1σ, and the coaction is determined by:uu1+a2x. Then there is a spectral sequenceE2=ExtˆF2[x]s,k+σ(F2,F2[a,u±1])π(ks)+σ(NF2)a. Explicitly, the completed Ext appearing in this E2-page may be calculated asE2=limnExtF2[x]/x2ns,k+σ(F2,F2[a,u±1]).

Corollary B

Let p and q denote integers such that p+q<0. Thenπp+qσ(NF2)a={0p0F2{aq}p=0.

Corollary B follows from straightforward computation of the Ext groups appearing in Theorem A. As we will explain in Section 5, it immediately implies Theorem 1.8 and hence Theorem 1.7.

Remark 1.9

Our proof of Theorem A arises by considering the descent spectral sequence for the C2-equivariant norm mapNF2F2_, where we use F2_ to denote the C2-equivariant Eilenberg–Maclane spectrum of the constant Mackey functor, HF2_. This norm map is a C2-equivariant refinement of the usual multiplication map F2F2F2, which arises from the fact that F2_ is a C2-commutative ring in C2-spectra in the sense of §2 (see, e.g., [21]). The basic descent datum is the RO(C2)-graded homotopy ofF2_NF2F2_, which is known as the Real topological Hochschild homology of F2_. These RO(C2)-graded homotopy groups were computed as an algebra in [8]. We will need to know them as a Hopf algebroid, and not just as an algebra. Our computation of the Hopf algebroid structure maps is likely of independent interest, and appears in Section 2.

Remark 1.10

By the Segal conjecture, and the fact that NF2 is 2-complete, the fixed points spectrum NF2C2 is identified with the more classical object (F2F2)hC2. There has been some interest in computing the homotopy groups of these fixed points, and we give a brief discussion in Section 6.

Outline

The spectral sequence in the main theorem is obtained by taking the a-completion of the relative Adams spectral sequence for the map NF2F2_. The E2-term of this spectral sequence is governed by the Hopf algebroid structure on π(F2_NF2F2_), otherwise known as Real topological Hochschild homology (cf. [8]). We determine this structure in §2 by comparison with underlying and geometric fixed points. In §3 we identify the E2-page for the Borel completion with the indicated limit of Ext groups. In §4 we compute these Ext groups, and extract a vanishing result which implies the Segal conjecture for the group C2. We give the proof of the Segal conjecture in §5. Finally, in §6 we indicate a computation of some low dimensional integer stems, and leave the reader with a few questions of interest.

Conventions

We assume the reader is acquainted with equivariant homotopy theory at the level of [11, §2,§3]. If (A,Γ) is a Hopf algebroid and M is a comodule, we will abbreviate ExtΓ(A,M) as ExtΓ(M).

Acknowledgments

We thank Danny Shi and Mingcong Zeng for discussions about their works in progress, and for their patience regarding our hasty and error-prone emails. We thank Hood Chatham for help with some computer calculations we used to explore our E2-page, as well as for his spectral sequence package. We thank J.D. Quigley and Tyler Lawson for sharing their unpublished work on the fixed points (NF2)C2, as well as for providing detailed answers to questions. We thank Mark Behrens for useful discussions, and for sharing a draft of his approach to the Segal conjecture via equivariant homotopy theory. Finally, we thank the anonymous referee for their careful reading and helpful comments. The first author was supported by the NSF under grant DMS-1803273, and the second author under grant DMS-1902669.

Section snippets

The real topological Hochschild homology of F2_

Let CAlgC2 denote the (∞-)category of C2-commutative algebras in genuine C2-spectra.1 Recall from [8] that, if RCAlgC2 is a C2-commutative ring spectrum,2

The construction of the spectral sequence

Theorem 3.1

There is a spectral sequenceE2=limnExtF2[x]/x2ns,k+σ(F2[a,u±1])π(ks)+σ(NF2)a.

Proof

Since NF2 is connective and 2-complete, there is an identification4:NF2holimΔF2_NF2+1. Since a-completion preserves homotopy limits, we have(NF2)aholimΔ(F2_NF2+1)a. Thus we get a spectral sequence with E1-term given byE1s,+s=π(F2_NF2s+1)a. Since F2_NF2F2_ is free as an F2_-module, the

Computation of the E2-page

In this section we compute some information about the E2-page of the spectral sequence from the previous section. Our principal aim will be to prove Corollary B from the Introduction.

Write {Er(n)} for the x-adic spectral sequence (as in [18, A1.3.9]):E1(n)=F2[a,u,y0,...,yn1]ExtF2[x]/(x2n)(F2[a,u]), where yi is represented by [x2i] in the cobar complex ([18, A1.2.11]), and write {Er} or {Er()} for the x-adic spectral sequenceE1=F2[a,u,yi:i0]ExtˆF2[x](F2[a,u]). These spectral sequences are

The Segal conjecture

In this section, we prove the Segal conjecture in the following form:

Theorem 5.1

Let X denote any bounded below spectrum. Then the natural mapNX(NX)a is an equivalence after 2-completion.

The key point is the following standard observation:

Lemma 5.2

Let X be a bounded below spectrum. Then, to prove Theorem 5.1, it suffices to show that(NX)[a1](NX)a[a1] is an equivalence after 2-completion. This in turn is equivalent to the claim that the Tate diagonalX(XX)tC2 is an equivalence after 2-completion.

Proof

The first

Epilogue

Integer stems

Over the last few years, there have been several attempts to understand the homotopy groups of the non-equivariant spectrum(NF2)C2=(F2F2)hC2. This seems especially interesting in light of forthcoming work of Mingcong Zeng and Lennart Meier, which uses the equivalenceΦC2NC2C4BPRNF2 to relate these homotopy groups to the slice spectral sequence differentials studied by Hill-Shi-Wang-Xu [12]

The most straightforward approach to these homotopy groups is via the homotopy fixed point

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