A homological approach to pseudoisotopy theory. I

Abstract

We construct a zig-zag from the once delooped space of pseudoisotopies of a closed $2n$-disc to the once looped algebraic $K$-theory space of the integers and show that the maps involved are $p$-locally $(2n-4)$-connected for $n>3$ and large primes $p$. The proof uses the computation of the stable homology of the moduli space of high-dimensional handlebodies due to Botvinnik--Perlmutter and is independent of the classical approach to pseudoisotopy theory based on Igusa's stability theorem and work of Waldhausen. Combined with a result of Randal-Williams, one consequence of this identification is a calculation of the rational homotopy groups of ${\mathrm{BDiff}}_{\partial }(D^{2n+1})$ in degrees up to $2n-5$.

46 pages, to appear in Inventiones Mathematicae

Cited by in corpus (5)

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• Diffeomorphisms of odd-dimensional discs, glued into a manifold
• On automorphisms of high-dimensional solid tori
• Relative H-Principle and Contact Geometry
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