On the non-existence of elements of Kervaire invariant one

Transcription

1 On the non-existence of elements of Kervaire invariant one Michael University of Virginia ICM Topology Session, Seoul, Korea, 2014

2 Poincaré Conjecture & Milnor s Question Milnor s Questions How many smooth structures are there on the n-sphere? Theorem (Poincaré Conjecture: Smale-Freedman-Perelman) If M is a homotopy n-sphere that is a manifold, then M is homeomorphic to S n.

3 Kervaire-Milnor 1963 Definition Let Θ n be the group of h-cobordism classes of homotopy n-spheres with addition connect sum. ψ n Have a map Θ n ψ n π s n /Im(J).

4 Pontryagin s Work Introduction Definition A framed n-manifold is an n-manifold with a continuous choice of basis for the normal vectors at every point {M n R n+k }/cobordism π n+k S k Framings on S k Im(J)

5 Pontryagin s Computations π s 0 = Z: π s 1 = Z/2Z:

6 Framed Surgery Introduction π s 2 : Pontryagin: framed surgery

7 Consequences Introduction ψ 2 not onto The map Θ 2 π s 2 /Im(J) is not surjective. Get a map µ: H n (M; Z) Z/2Z. If we can do surgery: 0, if we can t: 1.

8 Back to Ψ n Definition Let bp n+1 be the subset of Θ n of those spheres that bound parallelizable (frameable) manifolds. Theorem (Kervaire-Milnor) If n 2 mod 4, then there is an exact sequence Ψ 0 bp n+1 Θ n n πn/im(j) s 0. If n 2 mod 4, then there is an exact sequence Ψ 0 bp n+1 Θ n n πn/im(j) s Φn Z/2 bp n 0.

9 bp n+1 bp n+1 has a simple structure: it s finite cyclic! Theorem (Kervaire-Milnor) 1 n 0 mod 2 bp n+1 = 1 or 2 ) n 1 mod 4 2 2k 2 (2 2k 1 1)num( 4Bk n = 4k 1 > 3. k Theorem (Adams, Mahowald) 1 n 2, 4, 5, 6 mod 8, Im(J) = 2 ) n 0, 1 mod 8, denom( Bk n = 4k 1. 4k

10 Kervaire Problem Introduction Definition (Kervaire Invariant) If M is a framed (4k + 2)-manifold, then the Kervaire invariant Φ 4k+2 is the obstruction to surgery in the middle dimension. Kervaire Invariant One Problem Is there a smooth n-manifold of Kervaire invariant one?

11 Adams Spectral Sequence Adams Spectral Sequence There is a spectral sequence with ( E 2 = Ext A H (Y ), H (X) ) and converging to [X, Y ]. (Adem) Ext 1 (F 2, F 2 ) is generated by classes h i, i 0. h j survives the Adams SS if R 2j admits a division algebra structure: d 2 (h j ) = h 0 hj 1 2.

12 Browder s Reformulation Theorem (Browder 1969) 1 There are no smooth Kervaire invariant one manifolds in dimensions not of the form 2 j There is such a manifold in dimension 2 j+1 2 iff h 2 j survives the Adams spectral sequence. Classical Examples h1 2: S(C) S(C) h2 2 : S(H) S(H) h2 3 : S(O) S(O)

13 Adams Spectral Sequence

14 Adams Spectral Sequence

15 Previous Progress Introduction Theorem (Mahowald-Tangora) The class h4 2 survives the Adams SS. Theorem (Barratt-Jones-Mahowald) The class h5 2 survives the Adams SS.

16 Main Theorem Introduction Theorem (H.-Hopkins-Ravenel) For j 7, h 2 j does not survive the Adams SS. We produce a cohomology theory Ω ( ) such that 1 the cohomology theory detects the Kervaire classes, 2 Ω 2 (pt) = 0, and 3 Ω k+256 (X) = Ω k (X). We rigidify to a C 8 -equivariant spectrum Ω O : Ω = Ω C 8 O ΩhC 8 O.

17 Cohomology Theories Cohomology Theory satisfying {Topological Spaces} E {Graded Abelian Groups} 1 Homotopy Invariance: f g E (f ) = E (g) 2 Excision: X = B A C, then have a long exact sequence E n (X) E n (C) E n (B) E n (A) E n+1 (X)... Example 1 Singular cohomology 2 K -theory (vector bundles on X)

18 Spectra in Algebraic Topology Idea Spectra represent cohomology theories: E n (X) = [X, E n ] Spectrum A sequence of spaces E 1, E 2,... together with equivalences E n = ΩEn+1 = Maps(S 1, E n+1 ) 1 Singular homology: HZ n = K (Z, n) 2 K -Theory: KU 2n = Z BU, KU 2n 1 = U.

19 Equivariant Homotopy Equivariant Homotopy Homotopy theory for spaces with a G-action. For H G, have fixed points E H. There are spheres for every real representation. Example If G = Z/2, then we have S ρ 2 = C + and S 2. ( ) S ρ 2 {e} ( ) = S 2 S ρ C2 2 = R + = S 1.

20 What is our cohomology theory? 1 Begin with the bordism theory for (almost) complex manifolds: MU. Theorem For a finite index subgroup H G, there is a multiplicative functor : H Spectra G Spectra. N G H ( ) MU (C 8) = N C 8 C 2 MU: MU MU MU MU 2 Localize: Ω O = 1 MU (C 8).

21 Structure of MU (C 8) Introduction Slice Basics Gap Theorem: π 2 Ω = 0 HFP & Periodicity: π k+256 Ω = π k Ω Goal Want to compute the homotopy groups of the fixed points for spectra like Ω O. Start with Schubert cells: The Grassmanians Gr n (C k ) all have cells of the form C m For MU (C 8), we therefore see three kinds of representation spheres: 1 S kρ 8 2 Ind C 8 C 4 S kρ 4 3 Ind C 8 C 2 S kρ 2

22 Advantages of the Slice SS Slice Basics Gap Theorem: π 2 Ω = 0 HFP & Periodicity: π k+256 Ω = π k Ω

23 Advantages of the Slice SS Slice Basics Gap Theorem: π 2 Ω = 0 HFP & Periodicity: π k+256 Ω = π k Ω

24 Slice Filtration of MU (C 8) Slice Basics Gap Theorem: π 2 Ω = 0 HFP & Periodicity: π k+256 Ω = π k Ω Theorem There is a multiplicative filtration of MU (C8) with associated graded Ind C 8 H(p) Sk(p)ρ H(p) HZ. Corollary p P There is a spectral sequence E s,t 2 = p P p =t H H(p) t s ( S k(p)ρ H(p)+V ; Z ) = π t s V MU (C8).

25 Computing with Slices Slice Basics Gap Theorem: π 2 Ω = 0 HFP & Periodicity: π k+256 Ω = π k Ω Key Fact The E 2 -term can be computed from equivariant simple chain complexes. Cellular Chains for S ρ 4 1 C (S 3 ) {}}{ Z} {{ Z 2 } Z 4 Z 4 C (S 1 )

26 Gaps Introduction Slice Basics Gap Theorem: π 2 Ω = 0 HFP & Periodicity: π k+256 Ω = π k Ω Theorem For any non-trivial subgroup H of C 8 and for any induced sphere Ind C 8 K Skρ K with k Z, H K 2 (Skρ K ; Z) = 0 1 If k 0 or k < 2, then C 2 = 0 2 If k = 1, 2, in the relevant degrees, the complex is Z Z 2 by 1 (1, 1). Corollary For any k, π 2 Σ kρ 8MU (C 8) = 0.

27 Slice Basics Gap Theorem: π 2 Ω = 0 HFP & Periodicity: π k+256 Ω = π k Ω Homotopy Fixed Points & Periodicity Euler and Orientation Classes Homology of representation spheres is generated by Euler classes and orientation classes for representations. Theorem 1 The fixed and homotopy fixed points of Ω O agree if all Euler classes are nilpotent. 2 The homotopy of the homotopy fixed points of Ω O is periodic if some regular representation is orientable.