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

Abstract

Given a rank-2 hermitian bundle over a $3$–manifold that is nontrivial 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.