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

Stephen F. Sawin
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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 \cite{DW90}. In particular, the spin invariants constructed by Kirby and Melvin \cite{KM91} are shown to be identical to the invariants associated to $\mathrm{SO}(3).$ 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.

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Stephen F. Sawin. 2000-05-22. Invariants of Spin Three-Manifolds From Chern-Simons Theory and Finite-Dimensional Hopf Algebras. https://arxiv.org/abs/math/9910106

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