Finiteness of classifying spaces of relative diffeomorphism groups of 3–manifolds

Abstract

The main theorem shows that if $M$ is an irreducible compact connected orientable 3–manifold with non-empty boundary, then the classifying space $BDiff\left(Mrel\partial M\right)$ of the space of diffeomorphisms of $M$ which restrict to the identity map on $\partial M$ has the homotopy type of a finite aspherical CW–complex. This answers, for this class of manifolds, a question posed by M Kontsevich. The main theorem follows from a more precise result, which asserts that for these manifolds the mapping class group $\mathcal{\mathscr{H}}\left(Mrel\partial M\right)$ is built up as a sequence of extensions of free abelian groups and subgroups of finite index in relative mapping class groups of compact connected surfaces.