Let M be a compact, connected, orientable 3-manifold with non-empty boundary. In this talk, we study the classifying space for the diffeomorphism group of M fixing the boundary pointwise, and show that it has the homotopy type of a finite CW complex. This parallels analogous results of Gramain and Earle-Schatz for surfaces, and confirms a conjecture of Kontsevich for orientable 3-manifolds.

The proof will take us on a crash course in 3-manifold topology, and will feature a combination of results on geometrization of 3-manifolds with a topological poset parametrizing embedded spheres in M. This is joint work with Rachael Boyd and Jan Steinebrunner.