Mapping Class Groups of Compression Bodies and 3-Manifolds

We analyze the mapping class group Hx(W ) of automorphisms of the exterior boundary W of a compression body (Q,V ) of dimension 3 or 4 which extend over the compression body. Here V is the interior boundary of the compression body (Q,V ). Those automorphisms which extend as automorphisms of (Q,V ) rel V are called discrepant automorphisms, forming the mapping class group Hd(W ) of discrepant automorphisms of W in Q. If H(V ) denotes the mapping class group of V we describe a short exact sequence of mapping class groups relating Hd(W ), Hx(W ), and H(V ). For an orientable, compact, reducible 3-manifold W , there is a canonical “maximal” 4-dimensional compression body whose exterior boundary is W and whose interior boundary is the disjoint union of the irreducible summands of W . Using the canonical compression body Q, we show that the mapping class group H(W ) of W can be identified with Hx(W ). Thus we obtain a short exact sequence for the mapping class group of a 3-manifold, which gives the mapping class group of the disjoint union of irreducible summands as a quotient of the entire mapping class group by the group of discrepant automorphisms. The group of discrepant automorphisms is described in terms of generators, namely certain “slide” automorphisms, “spins,” and “Dehn twists” on 2-spheres. Much of the motivation for the results in this paper comes from a research program for classifying automorphisms of compact 3-manifolds, in the spirit of the Nielsen-Thurston classification of automorphisms of surfaces.