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On the Non-Existence of Elements of Kervaire Invariant ON THE NON-EXISTENCE OF ELEMENTS OF KERVAIRE INVARIANT ONE M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL Dedicated to Mark Mahowald 0 Abstract. We show that the Kervaire invariant one elements θj ∈ π2j+1−2S 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. Contents 1. Introduction 1 2. Equivariant stable homotopy theory 8 3. Mackey functors, homology and homotopy 35 4. The slice filtration 44 5. The complex cobordism spectrum 63 6. The Slice Theorem and the Reduction Theorem 76 7. The Reduction Theorem 80 8. The Gap Theorem 88 9. The Periodicity Theorem 89 10. The Homotopy Fixed Point Theorem 101 11. The Detection Theorem 103 Appendix A. The category of equivariant orthogonal spectra 118 Appendix B. Homotopy theory of equivariant orthogonal spectra 143 References 217 arXiv:0908.3724v4 [math.AT] 25 May 2015 1. Introduction The existence of smooth framed manifolds of Kervaire invariant one is one of the oldest unresolved issues in differential and algebraic topology. The question originated in the work of Pontryagin in the 1930’s. It took a definitive form in the paper [41] of Kervaire in which he constructed a combinatorial 10-manifold with no smooth structure, and in the work of Kervaire-Milnor [42] on h-cobordism classes 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. 1 2 M.A.HILL,M.J.HOPKINS,ANDD.C.RAVENEL of manifolds homeomorphic to a sphere. The question was connected to homotopy theory by Browder in his fundamental paper [13] where he showed that smooth framed manifolds of Kervaire invariant one exist only in dimensions of the form (2j+1 − 2), and that a manifold exists in that dimension if and only if the class 2 2,2j+1 hj ∈ ExtA (Z/2, Z/2) in the E2-term of the classical Adams spectral represents an element 0 θj ∈ π2j+1−2S 2 in the stable homotopy groups of spheres. The classes hj for j ≤ 3 represent the 0 squares of the Hopf maps. The element θ4 ∈ π30S had been observed in existing computations [56, 52, 60], and was constructed explicitly as a framed manifold by 0 Jones [39]. The element θ5 ∈ π60S was constructed by Barratt-Mahowald, and Barratt-Jones-Mahowald (see [9] and the discussion therein). The purpose of this paper is to prove the following theorem j+1 2 2,2 Theorem 1.1. For j ≥ 7 the class hj ∈ ExtA (Z/2, Z/2) does not represent an element of the stable homotopy groups of spheres. In other words, the Kervaire invariant elements θj do not exist for j ≥ 7. Smooth framed manifolds of Kervaire invariant one therefore exist only in di- mensions 2, 6, 14, 30, 62, and possibly 126. At the time of writing, our methods still leave open the existence of θ6. Many open issues in algebraic and differential topology depend on knowing whether or not the Kervaire invariant one elements θj exist for j ≥ 6. The fol- lowing results represent some of the issues now settled by Theorem 1.1. In the statements, the phrase “exceptional dimensions” refers to the dimensions 2, 6, 14, 30, 62, and 126. In all cases the situation in the dimension 126 is unresolved. By Browder’s work [13] the results listed below were known when the dimension in question was not 2 less than a power of 2. Modulo Browder’s result [13] the reduction of the statements to Theorem 1.1 can be found in the references cited. Theorem 1.2 ([42, 46]). Except in the six exceptional dimensions, every stably framed smooth manifold is framed cobordant to a homotopy sphere. In the first five of the exceptional dimensions it is known that not every sta- bly framed manifold is framed cobordant to a homotopy sphere. The situation is unresolved in dimension 126. Theorem 1.3 ([42]). Let M m be the manifold with boundary constructed by plumb- ing together two copies of the unit tangent bundle to S2k+1 (so m = 4k +2), and set Σm−1 = ∂M m. Unless m is one of the six exceptional dimensions, the space M m/Σm−1 is a triangulable manifold which does not admit any smooth structure, and the manifold Σm−1 (the Kervaire sphere) is homoemorphic but not diffeomor- phic to Sm−1. In the first five of the exceptional cases, the Kervaire sphere is known to be diffeomorphic to the ordinary sphere, and the Kervaire manifold can be smoothed. Theorem 1.4 ([42, 46]). Let Θn be the Kervaire-Milnor group of h-cobordism classes of homotopy n-spheres. Unless (4k + 2) is one of the six exceptional dimen- sions, 0 Θ4k+2 ≈ π4k+2S KERVAIRE INVARIANT ONE 3 and 0 |Θ4k+1| = ak π4k+1S , where ak is 1 if k is even, and 2 if k is odd. Theorem 1.5 ([8]). Unless n is 1, or one of the six exceptional dimensions, the n+1 Whitehead square [ιn+1, ιn+1] ∈ π2n+1S is not divisible by 2. 1.1. Outline of the argument. Our proof builds on the strategy used by the third author in [70] and on the homotopy theoretic refinement developed by the second author and Haynes Miller (see [73]). We construct a multiplicative cohomology theory Ω and establish the following results: 0 Theorem 1.6 (The Detection Theorem). If θj ∈ π2j+1−2S is an element of Ker- 2−2j+1 vaire invariant 1, and j > 2, then the “Hurewicz” image of θj in Ω (pt) is non-zero. Theorem 1.7 (The Periodicity Theorem). The cohomology theory Ω is 256-fold periodic: For all X, Ω∗(X) ≈ Ω∗+256(X). Theorem 1.8 (The Gap Theorem). The groups Ωi(pt) are zero for 0 <i< 4. These three results easily imply Theorem 1.1. The Periodicity Theorem and the Gap Theorem imply that the groups Ωi(pt) are zero for i ≡ 2 mod 256. By the 2−2j+1 Detection Theorem, if θj exists it has a non-zero Hurewicz image in Ω (pt). But this latter group is zero if j ≥ 7. 1.2. The cohomology theory Ω. Write Cn for the cyclic group of order n. Our cohomology theory Ω is part of a pair (Ω, ΩO) analogous to the orthogonal and unitary K-theory spectra KO and KU. The role of complex conjugation on KU is played by an action of C8 on ΩO, and Ω arises as its fixed points. It is better to think of ΩO as generalizing Atiyah’s C2-equivariant KR-theory [7], and in fact ΩO is constructed from the corresponding real bordism spectrum, as we now describe. Let MUR be the C2-equivariant real bordism spectrum of Landweber [43] and Fujii [24]. Roughly speaking one can think of MUR as describing the cobordism theory of real manifolds, which are stably almost complex manifolds equipped with a conjugate linear action of C2, such as the space of complex points of a smooth variety defined over R. A real manifold of real dimension 2n determines a homotopy class of maps nρ2 S → MUR where nρ2 is the direct sum of n copies of the real regular representation of C2, and Snρ2 is its one point compactification. Write ((C8)) MU = MUR ∧ MUR ∧ MUR ∧ MUR for the C8-equivariant spectrum gotten by smashing 4 copies of MUR together and letting C8 act by (a,b,c,d) 7→ (d,a,b,c¯ ). Very roughly speaking, MU ((C8)) can be thought of as the cobordism theory of stably almost manifolds equipped with a C8-action, with the property that the 4 M.A.HILL,M.J.HOPKINS,ANDD.C.RAVENEL restriction of the action to C2 ⊂ C8 determines a real structure. If M is a real manifold then M × M × M × M with the C8-action (a,b,c,d) 7→ (d,a,b,c¯ ) is an example. A suitable C8-manifold M of real dimension 8n determines a homo- topy class of maps Snρ8 → MU ((C8)), where nρ8 is the direct sum of n copies of the real regular representation of C8, and Snρ8 is its one point compactification. To define Ω we invert an equivariant analogue D : Sℓρ8 → MU ((C8)) −1 ((C8)) of the Bott periodicity class and form the C8-equivariant spectrum ΩO = D MU (in fact ℓ works out to be 19). The cohomology theory Ω is defined to be the ho- motopy fixed point spectrum of the C8-action on ΩO. There is some flexibility in the choice of D, but it needs to be chosen in order that the Periodicity Theorem holds, and in order that the map from the fixed point spectrum of ΩO to the homotopy fixed point spectrum be a weak equivalence. It also needs to be chosen in such a way that the Detection Theorem is preserved (see Remark 11.14). That such an D can be chosen with these properties is a relatively easy fact, albeit mildly technical. It is specified in Corollary 9.21. It can be described in the form M × M × M × M for a suitable real manifold M, though we do not do so.
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