Virtual Cohomological Dimension of Mapping Class Groups of 3-manifolds

The mapping class group of a topological space is the group of self-homeomorphisms modulo the equivalence relation of isotopy. For 2-manifolds (of finite type), it is a discrete group which is known (see [M, HI, H2, H3, H4]) to share many of the properties of arithmetic subgroups of linear algebraic groups, although it is not arithmetic. In this note we describe the results of [Ml], which show that the mapping class groups of many 3-manifolds share some of these properties. More precisely, a group G is said to be of type FL if the trivial G-module Z admits a resolution of finite length by finitely generated free G-modules (see [S]), and is said to be a duality group when there is a G-module C such that cap product with an element e G Hn (<?; C) induces isomorphisms H(G; A) S Hn-k(G; C®A) for all k and A (see [B-E]). The classes of groups of type FL and duality groups are closed under extension. The main result of [Ml] is