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.

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This paper represents an attempt, only partially successful, to get at what appear to be some deep and hitherto unexamined properties of the stable homotopy category. This work was motivated by the discovery of the pervasive manifestation of various types of periodicity in the E2-term of the Adams-Novikov spectral sequence converging to the stable homotopy groups of spheres. In section 3 of [34] and section 8 of [41], we introduced the chromatic spectral sequence, which converges to the above E2-term. Unlike most spectral sequences, its input is in some sense more interesting than its output, as the former displays many appealing patterns which are somewhat hidden in the latter (see section 8 of [41] for a more detailed discussion). It is not so much a computational aid (although it has been used [34] for computing the Novikov 2-line) as a conceptual tool for understanding certain qualitative aspects of the Novikov E2-term. Since the Novikov E2-term is a reasonably good approximation to stable homotopy itself, one is led to hope that the periodicity in it displayed by the chromatic spectral sequence is more than just an artifice of the algebra of complex cobordism. Hopefully, there is some sort of geometric periodicity behind the algebraic periodicity of the chromatic El-term. More specifically, we conjecture (5.8) that certain short exact sequences of BP*BPcomodules (5.6) used to construct the chromatic spectral sequence can be realized by cofibrations (5.7) and that the spectra involved enjoy a similar sort of periodicity (5.9). In attempting to prove this conjecture, we soon became aware of Bousfield's work on localization with respect to generalized homology, the relevant portions of which are described in section 1. For each generalized homology theory E*, Bousfield [10] defines an idempotent functor LE on the stable homotopy category whose image is equivalent to the category of fractions defined by Adams in section III 14 of [4]. If both X and E (the Manuscript received August 24, 1981; revised June 2, 1983. *Partially supported by N.S.F., S.R.C., and a Sloan Fellowship. 351 1.4. Definition. An E*-localization functor LE is a covariant functor from S, the stable homotopy category, to itself along with a natural transformationfrom the identify functor to LE such that tx: X-+ LEX is the terminal E*-equivalence (i.e., map inducing an isomorphism in E* (*)) fromX, i.e. (i) 71X X-+ LEX is an E-equivalence, and (ii) for any E*-equivalence f: X-k Y there is a unique r: Y-LEX such that rf = 77x. The following elementary results are left to the reader. 1.5. PROPOSITION. If the functor LE exists, (i) it is unique, (ii) it is idempotent, i.e. LELE = LE, and (iii) for any map g: X Ywhere Yis E*-local, there is a unique map g: LEX-Y such that #qx f.D 1.6. PROPOSITION. If LE exists and W-+ X-+ Y is a cofibre sequence, so is LE W-LEX-+ LE Y. 1.7. PROPOSITION. The homotopy inverse limit (see [12] Chapter XI or [4] p. 325) of E*-local spectra is E*-local. C

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Introduction. Of the many generalized homology theories available, very few are computable in practice except for the simplest of spaces. Standard homology and K-theory are the only ones which can be considered somewhat accessible. In recent years, complex cobordism, or equivalently, Brown-Peterson homology, has become a useful tool for algebraic topology. The high state of this development is particularly apparent with regard to BP stable operations, which are understood well enough to have many applications to stable homotopy; see for example [16]. Despite this achievement, it is still virtually impossible to compute the Brown-Peterson homology of any but the nicest of spaces; for example: some simple classifying spaces, spaces with no torsion and spaces with few cells. As a replacement for Brown-Peterson homology in this respect, we present the closely related generalized homologies known as the Morava K-theories. These are a sequence of homology theories, K(n) *(-), n > 0, for each prime p. The n = 1 case is essentially standard mod p complex K-theory. These theories are periodic of period 2(pn-1) and fit together to give Morava's beautiful structure theorem for complex cobordism; see [11]. Because of their close relationship to complex bordism, information about them will sometimes suffice for bordism, and thus geometric, problems. This is the case with our proof of the Conner-Floyd conjecture. The Morava K-theories each possess Kiunneth isomorphisms for all spaces. This feature enhances their computability tremendously. We demonstrate this point by computing the Morava K-theories of the Eilenberg-MacLane spaces. These spaces are difficult to handle even for Both authors are Alfred P. Sloan Research Fellows and partially supported by the N.S.F.

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...