Diffeomorphisms of odd-dimensional discs, glued into a manifold

TL;DR

This paper investigates how embedding diffeomorphisms of odd-dimensional discs into manifolds affects their rational homotopy and homology, revealing injectivity in homotopy and triviality in homology under certain conditions.

Contribution

It proves that the map from disc diffeomorphisms to manifold diffeomorphisms is injective on rational homotopy groups and trivial on rational homology when the manifold contains many specific embedded spheres.

Findings

Injective on rational homotopy groups

Trivial on rational homology under certain conditions

Relates to smooth torsion invariants and diffeomorphism theory

Abstract

For a compact $(2n+1)$-dimensional smooth manifold, let $\mu_M : B Diff_\partial (D^{2n+1}) \to B Diff (M)$ be the map that is 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 $ B Diff_\partial (D^{2n+1})$ are known in the concordance stable range. We prove two results on the behaviour of the map $\mu_M$ in the concordance stable range. Firstly, it is \emph{injective} on rational homotopy groups, and secondly, it is \emph{trivial} on rational homology, if $M$ contains sufficiently many embedded copies of $S^n\times S^{n+1} \setminus int(D^{2n+1})$. The homotopical statement is probably not new and follows from the theory of smooth torsion invariants. The homological statement relies on work by Botvinnik and Perlmutter on diffeomorphism of odd-dimensional manifolds.