Homology TQFT's and the Alexander-Reidemeister invariant of 3-manifolds via Hopf algebras and skein theory

摘要

We develop an explicit skein-theoretical algorithm to compute the Alexander polynomial of a 3-manifold from a surgery presentation employing the methods used in the construction of quantum invariants of 3-manifolds. As a prerequisite we establish and prove a rather unexpected equivalence between the topological quantum field theory constructed by Frohman and Nicas using the homology of U(1)-representation varieties on the one side and the combinatorially constructed Hennings TQFT based on the quasitriangular Hopf algebra N = Z/2 proportional toLambda* R-2 on the other side. We find that both TQFT's are SL(2, R)-equivariant functors and, as such, are isomorphic. The SL(2, R)-action in the Hennings construction comes from the natural action on X and in the case of the Frohman-Nicas theory from the Hard-Lefschetz decomposition of the U(1)-moduli spaces given that they are naturally Kahler. The irreducible components of this TQFT, corresponding to simple representations of SL(2, Z) and Sp(2g, Z), thus yield a large family of homological TQFT's by taking sums and products. We give several examples of TQFT's and invariants that appear to fit into this family, such as Milnor and Reidemeister Torsion, Seiberg-Witten theories, Casson type theories for homology circles A la Donaldson, higher rank gauge theories following Frohman and Nicas, and the Z/pZ reductions of Reshetikhin-Turaev theories over the cyclotomic integers Z[zeta(p)]. We also conjecture that the Hermings TQFT for quantum-sl(2) is the product of the Reshetikhin-Turaev TQFT and such a homological TQFT. [References: 52]