Moduli spaces of 3-manifolds with boundary are finite

TL;DR

This paper proves that the classifying space of the diffeomorphism group of any compact, orientable 3-manifold with boundary is homotopy equivalent to a finite CW complex, confirming a conjecture and advancing understanding of 3-manifold symmetries.

Contribution

The authors establish that B Diff(M rel boundary) is a finite CW complex for 3-manifolds with boundary, confirming a conjecture and showing B Diff(M) has finite type.

Findings

B Diff(M rel boundary) is homotopy equivalent to a finite CW complex.

B Diff(M) has finite type for all compact, orientable 3-manifolds.

Confirmed a conjecture by Kontsevich on the finiteness of classifying spaces.

Abstract

We study the classifying space B Diff(M) of the diffeomorphism group of a connected, compact, orientable 3-manifold M. In the case that M is reducible we build a contractible space parametrising the systems of reducing spheres. We use this to prove that if M has non-empty boundary, then B Diff(M rel boundary) has the homotopy type of a finite CW complex. This was conjectured by Kontsevich and appears on the Kirby problem list as Problem 3.48. As a consequence, we are able to show that for every compact, orientable 3-manifold M, B Diff(M) has finite type.