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.

Abstract. We study homotopy-coherent commutative multiplicative structures on equivariant spaces and spectra. We define N∞ operads, equivariant generalizations of E∞ operads. Algebras in equivariant spectra over an N∞ operad model homotopically commutative equivariant ring spectra that only admit certain collections of Hill-Hopkins-Ravenel norms, determined by the operad. Analogously, algebras in equivariant spaces over an N∞ operad provide explicit constructions of certain transfers. This characterization yields a conceptual explanation of the structure of equivariant infinite loop spaces.To explain the relationship between norms, transfers, and N∞ operads, we discuss the general features of these operads, linking their properties to families of finite sets with group actions and analyzing their behavior under norms and geometric fixed points. A surprising consequence of our study is that in stark contract to the classical setting, equivariantly the little disks and linear isometries operads for a general incomplete universe U need not determine the same algebras.Our work is motivated by the need to provide a framework to describe the flavors of commutativity seen in recent work of the second author and Hopkins on localization of equivariant commutative ring spectra.

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...

The cohomology theory known as Tmf, for "topological modular forms," is a universal object mapping out to elliptic cohomology theories, and its coefficient ring is closely connected to the classical ring of modular forms. We extend this to a functorial family of objects corresponding to elliptic curves with level structure and modular forms on them. Along the way, we produce a natural way to restrict to the cusps, providing multiplicative maps from Tmf with level structure to forms of K-theory. In particular, this allows us to construct a connective spectrum tmf0(3) consistent with properties suggested by Mahowald and Rezk. This is accomplished using the machinery of logarithmic structures. We construct a presheaf of locally even-periodic elliptic cohomology theories, equipped with highly structured multiplication, on the log-étale site of the moduli of elliptic curves. Evaluating this presheaf on modular curves produces Tmf with level structure.

We describe the slice spectral sequence of a 32-periodic C 4 -spectrum K [2] related to the C 4 norm