On the non-existence of elements of Kervaire invariant one

Abstract

We show that the Kervaire invariant one elements θ j ∈ π 2 j+1 −2 S 0 exist only for j ≤ 6. By Browder's Theorem, this means that smooth framed manifolds of Kervaire invariant one exist only in dimensions 2, 6, 14, 30, 62, and possibly 126. Except for dimension 126 this resolves a longstanding problem in algebraic topology. M. A. Hill was partially supported by NSF grants DMS-0905160 , DMS-1307896 and the Sloan foundation. M. J. Hopkins was partially supported the NSF grant DMS-0906194. D. C. Ravenel was partially supported by the NSF grants DMS-1307896 and DMS-0901560. All three authors received support from the DARPA grants HR0011-10-1-0054-DOD35CAP and FA9550-07-1-0555.11.6. Acknowledgments. First and foremost the authors would like to thank Ben Mann and the support of DARPA through the grant number FA9550-07-1-0555. It was the urging of Ben and the opportunity created by this funding that brought the authors together in collaboration in the first place. Though the results described in this paper were an unexpected outcome of our program, it's safe to say they would not have come into being without Ben's prodding. As it became clear that the techniques of equivariant homotopy theory were relevant to our project we drew heavily on the paper [35] of Po Hu and Igor Kriz. We'd like to acknowledge a debt of influence to that paper, and to thank the authors for writing it. We were also helped by the thesis of Dan Dugger (which appears as [20]). The second author would like to thank Dan Dugger, Marc Levine, Jacob Lurie, and Fabien Morel for several useful conversations. Early drafts of this manuscript were read by Mark Hovey, Tyler Lawson, and Peter Landweber, and the authors would like to express their gratitude for their many detailed comments. We also owe thanks to Haynes Miller for a very thoughtful and careful reading of our earlier drafts, and for his helpful suggestions for terminology. Thanks are due to Stefan Schwede for sharing with us his construction of M U R , to Mike Mandell for diligently manning the hotline for questions about the foundations of equivariant orthogonal spectra, to Andrew Blumberg for his many valuable comments on the second revision, and to Anna Marie Bohmann and Emily Riehl for valuable comments on our description of "working fiberwise."Finally, and most importantly, the authors would like to thank Mark Mahowald for a lifetime of mathematical ideas and inspiration, and for many helpful discussions in the early stages of this project. Equivariant stable homotopy theoryWe will work in the category of equivariant orthogonal spectra [54,53]. In this section we survey some of the main properties of the theory and establish some notation. The definitions, proofs, constructions, and other details are explained in Appendices A and B. The reader is also referred to the books of tom Dieck [80,79], and the survey of Greenlees and May [26] for an overview of equivariant stable homotopy theory, and for further references.We set up the basics of equivariant stable homotopy theory in the framework o...

Equivariant Structure on Smash Powers

“…Our method on the other hand allows us to recognize the diagonal map as a natural isomorphism, strengthening their statement. Note that the "Slice Cells" discussed in [HHR,4.1] are special cases of generating SI G -cofibrations in our language, and from this viewpoint the filtration given in [HHR,A.4.3] and our Theorem 3.2.14 achieve similar goals -an equivariant filtration of the smash power -with different methods. Finally note that the change of the indexing of the smash power away from a non discrete set we worked for in this section, is only addressed in the side note [HHR,A.35].…”

“…Note that the "Slice Cells" discussed in [HHR,4.1] are special cases of generating SI G -cofibrations in our language, and from this viewpoint the filtration given in [HHR,A.4.3] and our Theorem 3.2.14 achieve similar goals -an equivariant filtration of the smash power -with different methods. Finally note that the change of the indexing of the smash power away from a non discrete set we worked for in this section, is only addressed in the side note [HHR,A.35]. Since all groups discussed there are finite, this is not a major point in [HHR], but as we are going to move towards tori and more general compact Lie groups now, the details become important.…”

“…This proposition serves as a good starting point for calculating geometric fixed points of smash powers of general (H , H )-cofibrant orthogonal spectra. This has for example been studied in the special case of G a cyclic group by Kro in [Kr,3.10.1] and Hill, Hopkins and Ravenel in [HHR,B.96]. However, we are ultimately interested in smash powers of commutative ring spectra, and (H , H )-cofibrant approximation does not preserve strictly commutative ring spectra.…”

“…If one wants to start with an H-spectrum instead, this generalizes the study of multiplicative norm constructions, where X is assumed to be a discrete subgroup H of G. These norm functors have been famously put to use in the recent proof of the Kervaire-invariant problem by Hill, Hopkins and Ravenel. An introduction can be found in [HHR,A.3,4], or [S11, 8,9], both of which only became available very shortly before the third author's thesis (which very much is at the core of this paper) was finished. The third author learned about the interplay from Stefan Schwede, during a visit in Bonn in November 2010, where he presented the results of his thesis.…”

“…[HHR] address this problem by proving that the symmetric powers appearing in the generating q-cofibrations are "very flat" (cf. [HHR,B.13,63]), which allows them to construct a natural weak equivalence calculating geometric fixed points. Their method has the advantage that it is more easily applicable to the general multiplicative norm case they aim to study.…”