A homological approach to pseudoisotopy theory. I

DOI10.1007/S00222-021-01077-7arXiv2002.04647MaRDI QIDQ6334612

Manuel Krannich

Publication date: 11 February 2020

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 $mathrm{BDiff}_{p}artial(D^{2n+1})$ in degrees up to $2n-5$.

Mathematics Subject Classification ID

Surgery and handlebodies (57R65) Computations of higher (K)-theory of rings (19D50) Isotopy in differential topology (57R52) Infinite loop spaces (55P47)

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