Klein-four connections and the Casson invariant for non-trivial admissible $U(2)$ bundles

TL;DR

This paper explores the Casson invariant for non-trivial admissible U(2) bundles over 3-manifolds, revealing its divisibility properties linked to the mod 2 cohomology and Klein-four holonomy connections.

Contribution

It provides a new formula connecting the Casson invariant's divisibility to the mod 2 cohomology and Klein-four holonomy, extending understanding of flat connection counts.

Findings

The Casson invariant's 2-divisibility is governed by a cohomological formula.

The formula involves counting flat connections with Klein-four holonomy.

The work links topological invariants to specific flat connection structures.

Abstract

Given a rank 2 hermitian bundle over a 3-manifold that is non-trivial admissible in the sense of Floer, one defines its Casson invariant as half the signed count of its projectively flat connections, suitably perturbed. We show that the 2-divisibility of this integer invariant is controlled in part by a formula involving the mod 2 cohomology ring of the 3-manifold. This formula counts flat connections on the induced adjoint bundle with Klein-four holonomy.