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

Christopher Scaduto and Matthew Stoffregen

Algebraic & Geometric Topology 17 (2017) 2841–2861

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.

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Casson invariant, Lescop invariant, $2$–torsion
Mathematical Subject Classification 2010
Primary: 57M27
Received: 28 August 2016
Revised: 2 June 2017
Accepted: 20 June 2017
Published: 19 September 2017
Christopher Scaduto
Department of Mathematics
Brandeis University
Waltham, MA
United States
Matthew Stoffregen
Department of Mathematics
University of California
Los Angeles, CA
United States