Invariants for 1-dimensional cohomology classes arising from TQFT

Original · EN

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,χ) be a closed oriented n+1-manifold M with this extra structure together with χ∈ H¹(M). Let M∞ denote the infinite cyclic cover of M given by χ. Consider a fundamental domain E for the action of the integers on M∞ bounded by lifts of a surface Σ dual to χ, and in general position. E can be viewed as a cobordism from Σ 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 (Vₚ,Zₚ) theories of Blanchet,Habegger, Masbaum and Vogel when M is zero framed surgery to S³ 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 S³ along knots.