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Given a closed smooth manifold $M$ of even dimension $2n\ge6$ with finitefundamental group, we show that the classifying space ${\rm BDiff}(M)$ of thediffeomorphism group of $M$ is of finite type and has finitely generatedhomotopy groups in every degree. We also prove a variant of this result formanifolds with boundary and deduce that the space of smooth embeddings of acompact submanifold $N\subset M$ of arbitrary codimension into $M$ has finitelygenerated higher homotopy groups based at the inclusion, provided thefundamental group of the complement is finite. As an intermediate result, weshow that the group of homotopy classes of simple homotopy self-equivalences ofa finite CW complex with finite fundamental group is up to finite kernelcommensurable to an arithmetic group.

Finiteness properties of automorphism spaces of manifolds with finite fundamental group