The Arf-Kervaire problem in algebraic topology: Sketch of the proof

Abstract

Abstract. We provide a sketch of the proof of the non-existence of Kervaire Invariant one manifolds using equivariant homotopy theory [15].

Local Homotopy Theory

“…The formal part of this proof (along with the proofs of all of our other localization results) amounts to a reduction to the bounded monomorphism condition, which condition appears as Proposition 4.5.15 of [41].…”

“…The localization method which is presented here is not the most general, and has a handicrafted flavour by comparison to standard references such as [41], and more recent discussions of combinatorial model structures [5,75]. It does, however, have the benefit of being relatively simple to describe and apply in presheaf-theoretic contexts.…”

“…Aficionados of cellular model structures [41] will recognize that the injective model structure on the simplicial presheaf category sPre(C) is cellular, for set theoretic reasons.…”

“…What comes next can be expressed in terms of the modern theory of colocalizations [41], but for simplicity we use the language of Bousfield's original paper [11] on homology localizations.…”

Memorial Tribute: L. Gaunce Lewis Jr. (1949–2006) and Mark Steinberger (1950–2018)

“…The foundation text remains [11]." His praise of [11] in his review of an earlier expository paper [9] was still more effusive. 1 Gaunce's expertise, especially his remarkable application of Freyd's adjoint functor theorem to construct an adjunction between the prespectra and spectra of [11], is what made the original construction and analysis of the equivariant stable homotopy category possible.…”