Real topological Hochschild homology and the Segal conjecture
Introduction
The Segal conjecture, for the cyclic group of order 2, is an equivalence of the stable cohomotopy of with the completion of the Burnside ring of 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 , and for each integer let denote the Thom spectrum of . Then there is an equivalence of spectra
The only known proof of Lin's theorem proceeds via calculation of a certain continuous Ext group 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 . 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 rather than the Steenrod algebra . We trust the reader will agree that this reduces the complexity of the homological algebra.
Remark 1.2 Just as the Steenrod algebra arises as (the dual of) the homology of , the polynomial coalgebra appears as the topological Hochschild homology of .
To explain our methods, we must review how the Segal conjecture has been both restated and generalized via the language of -equivariant stable homotopy theory.
Notation 1.3 In the -equivariant stable homotopy category, we use the notation to denote the unit object. This is the -equivariant sphere spectrum. We use to denote the 1-point compactification of the sign representation. Depending on context, we use to denote either the field with 2 elements or the non-equivariant Eilenberg–Maclane spectrum .
Recollection 1.4 In the -equivariant stable homotopy category, the morphism is adjoint to the inclusion of fixed points into the sign representation σ. The Borel completion of a -spectrum X is the a-completion One says that a -spectrum X is Borel complete if the natural map is an equivalence.
Theorem 1.5 Lin's theorem, restated The natural map 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 is a -equivariant refinement of the smash product , with -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 map 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 . In other words, since is 2-complete, all statements of Lin's theorem are consequences of the following result:
Theorem 1.8 The -spectrum 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 and 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 , the virtual representation ring of . The functor Ext is then defined using relative injective resolutions in this category of -graded comodules, just as in [18, §A.1].
Theorem A Let be the Hopf algebra with x primitive of degree , and let be the comodule algebra where the class a is primitive in degree −σ, u is in degree , and the coaction is determined by: Then there is a spectral sequence Explicitly, the completed Ext appearing in this -page may be calculated as
Corollary B Let p and q denote integers such that . Then
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 -equivariant norm map where we use to denote the -equivariant Eilenberg–Maclane spectrum of the constant Mackey functor, . This norm map is a -equivariant refinement of the usual multiplication map , which arises from the fact that is a -commutative ring in -spectra in the sense of §2 (see, e.g., [21]). The basic descent datum is the -graded homotopy of which is known as the Real topological Hochschild homology of . These -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 is 2-complete, the fixed points spectrum is identified with the more classical object . 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 . The -term of this spectral sequence is governed by the Hopf algebroid structure on , 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 -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 . 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 is a Hopf algebroid and M is a comodule, we will abbreviate as .
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 -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 , 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
Let denote the (∞-)category of -commutative algebras in genuine -spectra.1 Recall from [8] that, if is a -commutative ring spectrum,2
The construction of the spectral sequence
Theorem 3.1 There is a spectral sequence
Proof Since is connective and 2-complete, there is an identification4: Since a-completion preserves homotopy limits, we have Thus we get a spectral sequence with -term given by Since is free as an -module, the
Computation of the -page
In this section we compute some information about the -page of the spectral sequence from the previous section. Our principal aim will be to prove Corollary B from the Introduction.
Write for the x-adic spectral sequence (as in [18, A1.3.9]): where is represented by in the cobar complex ([18, A1.2.11]), and write or for the x-adic spectral sequence 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 map 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 is an equivalence after 2-completion. This in turn is equivalent to the claim that the Tate diagonal 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 This seems especially interesting in light of forthcoming work of Mingcong Zeng and Lennart Meier, which uses the equivalence 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
References (22)
- et al.
The Segal conjecture for elementary Abelian p-groups
Topology
(1985) The localization of spectra with respect to homology
Topology
(1979)- et al.
Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence
Topology
(2001) On the localization of modules over the Steenrod algebra
J. Pure Appl. Algebra
(1980)- et al.
On the Adams spectral sequence for R-modules
Algebraic Geom. Topol.
(2001) - et al.
C2-equivariant stable homotopy from real motivic stable homotopy
- et al.
A -equivariant analog of Mahowald's Thom spectrum theorem
Proc. Am. Math. Soc.
(2018) - et al.
Ring Spectra and Their Applications
(1986) - et al.
On conjugation invariants in the dual Steenrod algebra
Proc. Am. Math. Soc.
(2000) - et al.
Real topological Hochschild homology
The -graded equivariant ordinary cohomology of complex projective spaces with linear actions
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