Invariants for 1-dimensional cohomology classes arising from TQFT

Abstract

Let $(V,Z)$ be a Topological Quantum Field Theory over a field $f$ defined on a cobordism category whose morphisms are oriented $n+1$-manifolds perhaps with extra structure. Let $(M,\chi)$ be a closed oriented $n+1$-manifold $M$ with this extra structure together with $\chi \in H^1(M).$ Let $M_{\infty}$ denote the infinite cyclic cover of $M$ given by $\chi.$ Consider a fundamental domain $E$ for the action of the integers on $M_{\infty}$ bounded by lifts of a surface $\Sigma$ dual to $\chi,$ and in general position. $E$ can be viewed as a cobordism from $\Sigma$ to itself. We give Turaev and Viro's proof of their theorem that the similarity class of the non-nilpotent part of $Z(E)$ is an invariant. We give a method to calculate this invariant for the $(V_p,Z_p)$ theories of Blanchet,Habegger, Masbaum and Vogel when $M$ is zero framed surgery to $S^3$ along a knot K. We give a formula for this invariant when $K$ is a twisted double of another knot. We obtain formulas for the quantum invariants of branched covers of knots, and unbranched covers of 0-surgery to $S^3$ along knots.