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