The Rohlin invariant and Z/2-valued invariants of homology spheres

TL;DR

This paper proves the uniqueness of the Rohlin invariant as a homomorphism on the Torelli group and extends the construction of homology 3-sphere invariants to include those with values in abelian groups with 2-torsion.

Contribution

It establishes the Rohlin invariant as the only invariant inducing a homomorphism on the Torelli group and generalizes the construction of homology sphere invariants to broader algebraic targets.

Findings

Rohlin invariant is unique in inducing a homomorphism on the Torelli group.

Extended invariants to include values in abelian groups with 2-torsion.

Provided a generalized framework for invariants of homology 3-spheres.

Abstract

In this paper we prove that the Rohlin invariant is the unique invariant inducing a homomorphism on the Torelli group. Using this result we generalize the construction of invariants of homology $3$-spheres from families of trivial 2-cocycles on the Torelli group given by Pitsch to include invariants with values on an abelian group with $2$-torsion.