On the nonexistence of elements of Kervaire invariant one

Abstract

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.

Gorenstein duality for real spectra

“…As explained in [16,Section 4.2], this leads to the definition of X ! P k X , which is the universal map into a C 2 -spectrum that is Ä k .…”

Equivariant Anderson Duality and Mackey Functor Duality

“…The slice filtration and the slice spectral sequence. We now recall the definition of the slice filtration and the slice spectral sequence from [HHR14]. However, since our focus is on the group Z/2, the exposition is easier.…”

“…Our main tool here is the slice spectral sequence of [HHR14] and the various differentials which have been determined by [HK01].…”

“…Even though the hypothesis of the last proposition could seem restrictive, the class of Z/2-spectra such that 2a acts as 0 contains very fundamental examples, the motivating class for our purpose are M R-modules, since 2a = 0 in M R −α (this is a consequence of the computations of [HHR14] by the slice spectral sequence).…”

The Kervaire invariant problem

“…It has been one of the oldest open issues in algebraic topology. My aim in these lectures is to describe the origins and history of this famous problem, and following recent result of Mike Hill, myself, and Doug Ravenel [11]. …”