  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.
