Based on work of Kervaire-Milnor, Browder, and Hill-Hopkins-Ravenel, Guozhen Wang and I recently proved that the 61-sphere has a unique smooth structure, and it is the last odd dimensional case: The only odd dimensional spheres that have a unique smooth structure are the ones in dimensions 1, 3, 5 and 61. The proof is a computation of stable homotopy groups of spheres. In recent work with Dan Isaksen and Guozhen Wang, we developed a new method that can be used to compute stable homotopy groups of spheres in a much more effective way than known ones. This new method uses motivic homotopy theory, which was introduced by Morel and Voevodsky to study algebraic geometry.
Smooth structures on spheres, stable homotopy groups of spheres, and motivic homotopy theory
We show that Lubin-Tate spectra at the prime are Real oriented and Real Landweber exact. The proof is an application of the Goerss--Hopkins--Miller theorem to algebras with involution. For each height , we compute the entire homotopy fixed point spectral sequence for with its -action by the formal inverse. We study, as the height varies, the Hurewicz images of the stable homotopy groups of spheres in the homotopy of these -fixed points. Then, I will talk about the slice spectral sequence of a -equivariant spectrum. This spectrum is a variant of the detection spectrum of Hill--Hopkins--Ravenel and is very closely related to the height 4 Lubin--Tate theory.
A fundamental problem in 4-dimensional topology is the following geography question: "which simply connected topological 4-manifolds admit a smooth structure?" After the celebrated work of Kirby-Siebenmann, Freedman, and Donaldson, the last uncharted territory of this geography question is the "11/8-Conjecture''. This conjecture, proposed by Matsumoto, states that for any smooth spin 4-manifold, the ratio of its second-Betti number and signature is least 11/8.
Furuta proved the ''10/8+2''-Theorem by studying the existence of certain Pin(2)-equivariant stable maps between representation spheres. In this talk, we will present a complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald invariants. In particular, we improve Furuta's result into a ''10/8+4''-Theorem. Furthermore, we show that within the current existing framework, this is the limit. This is joint work with Mike Hopkins, Jianfeng Lin and XiaoLin Danny Shi.
Smooth structures on spheres, stable homotopy groups of spheres, and motivic homotopy theory