Diffeomorphisms of odd-dimensional discs, glued into a manifold

Ebert, Johannes

doi:10.2140/agt.2023.23.2329

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Diffeomorphisms of odd-dimensional discs, glued into a manifold

Johannes Ebert

Algebraic & Geometric Topology 23 (2023) 2329–2345

DOI: 10.2140/agt.2023.23.2329

Abstract

Let $μ_{M} : {BDiff}_{\partial} (D^{2 n + 1}) \to BDiff (M)$ , for a compact $(2 n + 1)$ –dimensional smooth manifold $M$ , be the map defined by extending diffeomorphisms on an embedded disc by the identity. By a classical result of Farrell and Hsiang, the rational homotopy groups and the rational homology of ${BDiff}_{\partial} (D^{2 n + 1})$ are known in the concordance stable range. We prove two results on the behaviour of the map $μ_{M}$ in the concordance stable range. Firstly, it is injective on rational homotopy groups, and secondly, it is trivial on rational homology if $M$ contains sufficiently many embedded copies of $S^{n} \times S^{n + 1} ∖ int (D^{2 n + 1})$ . We also show that $μ_{M}$ is generally not injective on homotopy groups outside the stable range.

The homotopical statement is probably not new and follows from the theory of smooth torsion invariants. The noninjectivity outside the stable range is based on recent work by Krannich and Randal-Williams. The homological statement relies on work by Botvinnik and Perlmutter on diffeomorphisms of odd-dimensional manifolds.

Keywords

diffeomorphism groups of high-dimensional manifolds, smooth torsion invariants

Mathematical Subject Classification

Primary: 57S05

References

Publication

Received: 19 August 2021

Revised: 14 December 2021

Accepted: 26 December 2021

Published: 25 July 2023

Authors

Johannes Ebert
Fachbereich Mathematik und Informatik Westfälische Wilhelms-Universität Münster Münster Germany

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Volume 23, issue 5 (2023)