On the finiteness of the classifying space of diffeomorphisms of reducible three manifolds

TL;DR

This paper proves a homological version of Kontsevich's conjecture, showing that the classifying space of diffeomorphisms of certain reducible 3-manifolds has finitely generated homology groups, extending previous results for irreducible cases.

Contribution

It establishes that the classifying space of diffeomorphisms for reducible 3-manifolds has finitely many nonzero, finitely generated homology groups, generalizing prior irreducible manifold results.

Findings

Finitely many nonzero homology groups for the classifying space

Each homology group is finitely generated

Extension of results from irreducible to reducible 3-manifolds

Abstract

Kontsevich conjectured that $\text{BDiff}(M, \text{rel }\partial)$ has the homotopy type of a finite CW complex for all compact $3$-manifolds with non-empty boundary. Hatcher-McCullough proved this conjecture when $M$ is irreducible. We prove a homological version of Kontsevich's conjecture. More precisely, we show that $\text{BDiff}(M, \text{rel }\partial)$ has finitely many nonzero homology groups, each finitely generated, when $M$ is a connected sum of irreducible $3$-manifolds that each have a nontrivial and non-spherical boundary.