A homological approach to pseudoisotopy theory. I

Abstract

AbstractWe 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 $$(2n-4)$$ ( 2 n - 4 ) -connected for $$n\,{>}\,3$$ 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 $$\mathrm {BDiff}_\partial (D^{2n+1})$$ BDiff ∂ ( D 2 n + 1 ) in degrees up to $$2n-5$$ 2 n - 5 .