Homology TQFT's and the Alexander-Reidemeister Invariant of 3-Manifolds via Hopf Algebras and Skein Theory

Abstract

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 intersection homology of U(1)-representation varieties on the one side and the combinatorially constructed Hennings-TQFT based on the quasitriangular Hopf algebra ${\cal N}=\Z/2\ltimes \ext *\R^2$ on the other side. We find that both TQFT's are $SL(2,\mathbb{R})$-equivariant functors and also as such isomorphic. The $SL(2,\mathbb{R})$-action in the Hennings construction comes from the natural action on $\mathcal{N}$ 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 Kähler. The irreducible components of this TQFT, corresponding to simple representations of $SL(2,\mathbb{Z})$ and $Sp(2g,\mathbb{Z})$, 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 á la Donaldson, higher rank gauge theories following Frohman and Nicas, and the $\mathbb{Z}/r$ reductions of Reshetikhin-Turaev theories over the cyclotomic integers $\mathbb{Z}[{\zeta }_r]$. We also conjecture that the Hennings TQFT for quantum-${\mathfrak{s}l}_2$ is the product of the Reshetikhin-Turaev TQFT and such a homological TQFT.

50 pages 38 figures. Added chapters on Reidemeister torsion and skein theory, as well as new examples. Final revisions. To appear Canad. J. Math

References in corpus (6)

• Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity
• S and T matrices for the super $U(1,1)$ WZW model. Application to surgery and 3-manifold invariants based on the Alexander Conway polynomial
• Mapping Class Group Actions on Quantum Doubles
• Genealogy of Nonperturbative Quantum-Invariants of 3-Manifolds: The Surgical Family
• On the Connectivity of Cobordisms and Half-Projective TQFT's
• Resolutions of p-Modular TQFT's and Representations of Symmetric Groups