Topological Quantum Field Theories Associated to Finite Groups and Crossed G-Sets

Using methods suggested by the work of Turaev and Viro 11, 12, we provide a detailed construction of topological quantum field theories associated to finite crossed G-sets. Our construction of theories associated to finite groups fills in some details implicit in Dijkgraaf and Witten's 3 discussion of topological gauge theories with finite gauge group, while the theories associated to finite crossed G-sets simultaneously extend Dijkgraaf and Witten's 3 results to 3-manifolds equipped with links and Freyd and Yetter's 5 construction of link invariants from crossed G-sets from links in the 3-sphere to links in arbitrary 3-manifolds. Topological interpretations of the manifold and link invariants associated to these TQFT's are provided. We conclude discussion of our results as a toy model for QFT and of their relation to quantum groups.