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.
Categories: math.AT