Invariants of Spin Three-Manifolds From Chern–Simons Theory and Finite-Dimensional Hopf Algebras

Abstract

A version of Kirby calculus for spin and framed three-manifolds is given and is used to construct invariants of spin and framed three-manifolds in two situations. The first is ribbon *-categories which possess odd degenerate objects. This case includes the quantum group situations corresponding to the half-integer level Chern–Simons theories conjectured to give spin TQFTs by Dijkgraaf and Witten (1990, Commun. Math. Phys.129, 393–429). In particular, the spin invariants constructed by Kirby and Melvin (1991, Invent. Math.105, 473–545) are shown to be identical to the invariants associated to SO(3). Second, an invariant of spin manifolds analogous to the Hennings invariant is constructed beginning with an arbitrary factorizable, unimodular quasitriangular Hopf algebra. In particular a framed manifold invariant is associated to every finite-dimensional Hopf algebra via its quantum double, and is conjectured to be identical to Kuperberg's noninvolutory invariant of framed manifolds associated to that Hopf algebra.