Some finiteness results for groups of automorphisms of manifolds

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.

arXiv comment44 pages, 2 figures. To appear in Geometry & Topology. Corrected grant information

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Cited by in corpus (13)

• Mapping class groups of highly connected $(4k+2)$-manifolds
• On the cohomology of Torelli groups
• The cohomology of Torelli groups is algebraic
• Dalian notes on rational Pontryagin classes
• Homotopy types of diffeomorphism groups of polar Morse-Bott foliations on lens spaces, 1
• Embedding calculus and smooth structures
• Homotopy types of diffeomorphisms groups of simplest Morse-Bott foliations on lens spaces, 2
• Algebraic independence of topological Pontryagin classes
• On automorphisms of high-dimensional solid tori
• An Application of the $h$-principle to Manifold Calculus
• The homology of moduli spaces of 4-manifolds may be infinitely generated
• On the finiteness of the classifying space of diffeomorphisms of reducible three manifolds
• Some rational homology computations for diffeomorphisms of odd-dimensional manifolds