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Group categories and their field theories

Frank Quinn

Geometry & Topology Monographs 2 (1999) 407–453

DOI: 10.2140/gtm.1999.2.407

arXiv: math.GT/9811047


A group–category is an additively semisimple category with a monoidal product structure in which the simple objects are invertible. For example in the category of representations of a group, 1–dimensional representations are the invertible simple objects. This paper gives a detailed exploration of “topological quantum field theories” for group–categories, in hopes of finding clues to a better understanding of the general situation. Group–categories are classified in several ways extending results of Frölich and Kerler. Topological field theories based on homology and cohomology are constructed, and these are shown to include theories obtained from group–categories by Reshetikhin–Turaev constructions. Braided–commutative categories most naturally give theories on 4–manifold thickenings of 2–complexes; the usual 3–manifold theories are obtained from these by normalizing them (using results of Kirby) to depend mostly on the boundary of the thickening. This is worked out for group–categories, and in particular we determine when the normalization is possible and when it is not.

Dedicated to Rob Kirby, on the occasion of his 60th birthday.


topological quantum field theory, braided category

Mathematical Subject Classification

Primary: 18D10

Secondary: 55B20, 81R50


Received: 9 November 1998
Revised: 27 January 1999
Published: 21 November 1999

Frank Quinn
Department of Mathematics
Virginia Tech
Blacksburg VA 24061-0123