We show that although the mapping class group of a 3-manifold $W$ certainly fails to respect the prime decomposition of the 3-manifold, one can understand the failure by obtaining a relationship between the mapping class group of the manifold and the mapping class groups of the irreducible summands. In addition, we conjecture a relationship with the mapping class group of a maximal connected sum of $S2\times S1$’s in the prime decomposition.
Our method involves the use of a a canonical “maximal” 4-dimensional compression body whose exterior boundary is $W$ and whose interior boundary is the disjoint union $V$ of the irreducible summands of $W$. Each automorphism of $W$ extends over the 4-dimensional compression body and uniquely determines an automorphism of $V$. Letting ${\cal H}(W)$ denote the mapping class group of $W$, we obtain a short exact sequence $1\to {\cal H}_d(W)\to {\cal H}(W) \to {\cal H}(V)\to 1$, the first group being a mapping class group of “discrepant” automorphisms of $W$.
The motivation for this work comes from a program for understanding automorphisms of compact 3-manifolds by decomposing on surfaces invariant up to isotopy. This works well for irreducible manifolds, including those with compressible boundary, but often fails for reducible manifolds.