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A homological approach to pseudoisotopy theory. I
OAI: oai:www.repository.cam.ac.uk:1810/334326 • DOI: 10.17863/CAM.81739
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 $2n-5$.