Finiteness properties of automorphism spaces of manifolds with finite fundamental group

Abstract Given a closed smooth manifold M of even dimension $$2n\ge 6$$ 2 n ≥ 6 with finite fundamental group, we show that the classifying space $$B\textrm{Diff}(M)$$ B Diff ( M ) diffeomorphism group is type and has finitely generated homotopy groups in every degree. We also prove variant this result for manifolds boundary deduce embeddings compact submanifold $$N\subset M$$ N ⊂ arbitrary codimension into higher based at inclusion, provided complement finite. As an intermediate result, classes simple self-equivalences CW complex up to kernel commensurable arithmetic group.