Some finiteness results for groups of automorphisms of manifolds

TL;DR

This paper establishes finiteness properties for the homology and homotopy groups of classifying spaces of diffeomorphism groups of disks and related automorphism groups in certain dimensions, using advanced algebraic topology techniques.

Contribution

It proves finiteness of homology and homotopy groups for these classifying spaces in all but dimensions 4, 5, and 7, extending understanding of automorphism groups of manifolds.

Findings

Homology and homotopy groups are finitely generated in specified dimensions.

Results apply to homeomorphisms of ^n and automorphisms of 2-connected manifolds.

Uses homological stability, embedding calculus, and arithmeticity of mapping class groups.

Abstract

We prove that in dimensions not equal to 4, 5, or 7, the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely generated in each degree. The proof uses homological stability, embedding calculus and the arithmeticity of mapping class groups. From this we deduce similar results for the homeomorphisms of $R^n$ and various types of automorphisms of 2-connected manifolds.