Diffeomorphisms of odd-dimensional discs, glued into a manifold

Abstract

For a compact $(2n+1)$-dimensional smooth manifold, let ${\mu }_M:BDif{f}_{\partial }(D^{2n+1})\to BDiff(M)$ 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 $BDif{f}_{\partial }(D^{2n+1})$ 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.

Major revision and added results. Final version, to appear in Algebraic and Geometric Topology

Cited by in corpus (2)

• Relative H-Principle and Contact Geometry
• Some rational homology computations for diffeomorphisms of odd-dimensional manifolds