Finiteness properties of automorphism spaces of manifolds with finite fundamental group.

Given a closed smooth manifold $M$ of even dimension $2n\ge6$ with finite fundamental group, we show that the classifying space ${\rm BDiff}(M)$ of the diffeomorphism group of $M$ is of finite type and has finitely generated homotopy groups in every degree. We also prove a variant of this result for manifolds with boundary and deduce that the space of smooth embeddings of a compact submanifold $N\subset M$ of arbitrary codimension into $M$ has finitely generated higher homotopy groups based at the inclusion, provided the fundamental group of the complement is finite.