A homological approach to pseudoisotopy theory. I

Abstract

MANUEL KRANNICH A. We construct a zig-zag from the 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 BDiff ∂ (D 2n+1 ) in degrees up to 2n − 5.

Diffeomorphisms of odd-dimensional discs, glued into a manifold

“…This follows from the stable h-cobordism theorem [19] and Igusa's stability theorem [9]. An improvement of the range has recently been obtained by Krannich [15]; Corollary B of that paper shows that (1.1) holds whenever k ≤ 2n − 6 and n ≥ 4.…”

“…Remark 2.8. Krannich announced a follow-up to [15] which should give an even more direct proof of Theorem 2.7 that also improves the range of degrees up to 2n − 5. The existence of a rationally 2n [15], and the only remaining issue is the identification of this map with τ D 2n+1 , followed by the obvious rational equivalence hofib(Q…”

Homological stability: a tool for computations

“…This has since been used to compute e.g. ; homotopy groups of the diffeomorphisms of discs, or give a totally new approach to pseudoisotopy theory [40,42].…”