Finiteness properties of automorphism spaces of manifolds with finite fundamental group

TL;DR

This paper proves that the classifying space of the diffeomorphism group of certain finite fundamental group manifolds has finite type and finitely generated homotopy groups, extending to embeddings and related automorphism groups.

Contribution

It establishes finiteness properties of automorphism spaces of manifolds with finite fundamental group, including classifying spaces and embedding spaces, and relates automorphism groups to arithmetic groups.

Findings

Classifying space ${\rm BDiff}(M)$ has finite type and finitely generated homotopy groups.

Space of smooth embeddings of submanifolds has finitely generated higher homotopy groups.

Homotopy classes of simple homotopy self-equivalences are commensurable to an arithmetic group.

Abstract

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. As an intermediate result, we show that the group of homotopy classes of simple homotopy self-equivalences of a finite CW complex with finite fundamental group is up to finite kernel commensurable to an arithmetic group.