On the Connectivity of Cobordisms and Half-Projective TQFT's

We consider a generalization of the axioms of a TQFT, the so-called half-projective TQFT's, where we inserted an anomaly, , in the composition law. Here μ₀ is a coboundary (in a group cohomological sense) on the cobordism categories with non-negative, integer values. The element of the ring over which the TQFT is defined does not have to be invertible. In particular, it may be zero. This modification makes it possible to extend quantum-invariants, which vanish on S ¹×S ², to non-trivial TQFT's. Note, that a TQFT in the ordinary sense of Atiyah with this property has to be trivial all together. We organize our discussions such that the notion of a half-projective TQFT is extracted as the only possible generalization under a few very natural assumptions. Based on separate work with Lyubashenko on connected TQFT's, we construct a large class of half-projective TQFT's with . Their invariants all vanish on S ¹×S ², and they coincide with the Hennings invariant for non-semisimple Hopf algebras and, more generally, with the Lyubashenko invariant for non-semisimple categories. We also develop a few topological tools that allow us to determine the cocycle μ₀ and find numbers, ϱ(M), such that the linear map associated to a cobordism, M, is of the form . They are concerned with connectivity properties of cobordisms, as for example maximal non-separating surfaces. We introduce in particular the notions of “interior” homotopy and homology groups, and of coordinate graphs, which are functions on cobordisms with values in the morphisms of a graph category. For applications we will prove that half-projective TQFT's with vanish on cobordisms with infinite interior homology, and we argue that the order of divergence of the TQFT on a cobordism, M, in the “classical limit” can be estimated by the rank of its maximal free interior group, which coincides with ϱ(M). Received: 20 October 1997 / Accepted: 18 March 1998