A surgery approach to abelian quotients of the level 2 congruence group and the Torelli group

We provide algorithms for computing the Rochlin invariants of mod 2 homology spheres and mapping tori. This provides a unified framework for studying two families of maps: The Birman-Craggs maps of the Torelli group, and Sato’s maps of the level 2 congruence subgroup of the mapping class group. Our framework gives new, elementary proofs that both families of maps are homomorphisms, gives an explicit method for evaluating these maps on Dehn twists, and relates the two families when restricted to the Torelli group. It also gives a relation between an extension of the Birman–Craggs maps to the level 2 congruence subgroup, and Meyer’s signature cocycle. Our methods involve 3-manifold techniques, and do not depend on results in 4-manifold theory as in the original constructions.