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Talk

May 12th 2020
Manuel Krannich (University of Cambridge, UK): The pseudo-isotopy stable range

Abstract

By the s-cobordism theorem, isomorphism classes of h-cobordisms on a high-dimensional d-manifold M are in canonical bijection with a certain quotient of the first algebraic K-theory group of the integral group ring of π₁M—the Whitehead group. This has a parametrised refinement: work of Waldhausen provides a canonical map τ_M : H(M) → ΩWh(M) from the moduli space of h-cobordisms to a certain “quotient“ of the once looped algebraic K-theory spectrum of the spherical group ring of ΩM—the Whitehead spectrum.
From this point of view, the s-cobordism theorem says that τ_M is a bijection on components, so one might ask to which extent this generalises to the parametrised setting, i.e. how connected τ_M is. Combining the stable parametrised h-cobordism theorem with work of Igusa, it has been known for some time that τ_M is at least d/3-connected, but the optimal connectivity range remained subject to speculation, even up to a constant.
In this talk, I will explain a new perspective on this question and survey recent progress resulting from works of Kupers, Randal-Williams, and myself (in various combinations and partly ongoing).

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