  The main theorem shows that if M is an irreducible compact connected
orientable 3-manifold with non-empty boundary, then the classifying space
BDiff(M rel dM) of the space of diffeomorphisms of M which restrict to the
identity map on boundary(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 H(M rel dM) 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.
