Document Type

Article

Article Version

Post-print

Publication Date

2002

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.

Comments

NOTICE: this is the author’s version of a work that was accepted for publication in Advances in Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Advances in Mathematics [165, 1, 2002] DOI: 10.1006/aima.2000.1935

Publication Title

Advances in Mathematics

Published Citation

Sawin, Stephen F. 2002. Invariants of spin three-manifolds from Chern-Simons theory and finite-dimensional Hopf algebras. Advances in Mathematics 165 (1), 35-70.

DOI

10.1006/aima.2000.1935

Peer Reviewed

Download

Share

COinS