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

Scaduto, Christopher; Stoffregen, Matthew

doi:10.2140/agt.2017.17.2841

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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

DOI: 10.2140/agt.2017.17.2841

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.

Keywords

Casson invariant, Lescop invariant, $2$–torsion

Mathematical Subject Classification 2010

Primary: 57M27

References

Publication

Received: 28 August 2016

Revised: 2 June 2017

Accepted: 20 June 2017

Published: 19 September 2017

Authors

Christopher Scaduto
Department of Mathematics Brandeis University Waltham, MA United States

Matthew Stoffregen
Department of Mathematics University of California Los Angeles, CA United States

Volume 17, issue 5 (2017)