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

TL;DR

This paper introduces algorithms to compute Rochlin invariants, unifying the study of Birman-Craggs and Sato maps, and provides new proofs and explicit evaluation methods for these homomorphisms.

Contribution

It develops a unified framework for computing and relating the Birman-Craggs and Sato maps, offering elementary proofs and explicit evaluation techniques.

Findings

Algorithms for Rochlin invariant computation

Unified framework for Birman-Craggs and Sato maps

Explicit evaluation methods for Dehn twists

Abstract

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.