Invariants for one-dimensional cohomology classes arising from TQFT

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 (for example a p1 structure and banded link). Let (M, χ) be a closed oriented n + 1-manifold M with this extra structure together with χ ϵ H1(M). Suppose χ: H1(M) → Z is an epimorphism. 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 Viroʹs proof of their theorem that the similarity class of the nonnilpotent 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 0-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. We study periodicity among the quantum invariants of Brieskorn manifolds. We give an upper bound on the quantum invariants of branched covers of fibered knots. We also define finer invariants for pairs (M, χ) for TQFTs over Dedekind domains. We use these ideas to study isotopy invariants of banded links in S1 × S2.