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.

38 pages, in AmS-TeX- Version 2.1 , needs epsf.tex, decode figures.uu file to get fig1.eps through fig10.eps. This version corrects and improves the discussion of when p is ordinary with respect to n (on page 15), updates references, and includes other minor changes

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