Homologically Twisted Invariants Related to (2+1)- and (3+1)-Dimensional State-Sum Topological Quantum Field Theories

We outline a general construction applicable to the Turaev/Viro [TV], Crane/Yetter [CY] and generalized Turaev/Viro invariants (cf. [Y1]) of invariants valued in complex-valued functions on HD−2(M , GrC), where GrC is the abelian group of functorial tensor automorphisms on the artinian tortile category used to construct the TQFT. Introduction It is the purpose of this note to introduce a construction of invariants of 3and 4-manifolds which combines state-sum techniques (cf. [TV,CY,Y1,Y2]) with a dependence on (co)homology classes on the manifold. The basic ingredient is an extra piece of structure on the tensor categories from which the state-sum invariants are constructed: a (necessarily abelian) group of functorial tensor automorphisms, which is used as the coefficient group for homology. We assume a familiarity with standard references on monoidal and tensor categories, e.g. Mac Lane [M], Saavedra-Rivano [S], Kerler [Ke]. Throughout all manifolds are assumed to be piece-wise linear (or, equivalently, smooth). A note on terminology: thoughout, we adopt the convention of using names originally used by categorists (e.g. monoidal, autonomous, natural automorphism of the identity functor...) to refer to not-necessarily abelian categories with a given structure, and names originally used by algebraic geometers (e.g. tensor, rigid, functorial automorphism,...) to refer to abelian categories with a given structure implemented by (multi)linear functors exact in all variables. We advocate the adoption of this custom by other authors as a way of cutting down the profusion of terminology now afflicting quantum topology. This work was inspired by unpublished work of Paolo Cotta-Ramusino at the physical level of rigor, and by conversations with Cotta-Ramusino and Louis Kauffman at the XXII International Conference on Differential Geometric Methods in Theoretical Physics, Ixtapa, Mexico. The results were actually obtained while the author was in Ixtapa, so he extends thanks also to the organizers of the conference for their hospitality and financial support. Functorial Tensor Automorphisms As noted in the introduction, the basic ingredient in this construction is an extra piece of categorical structure available on the categories from which the Crane/Yetter [CY] and generalized Turaev/Viro [Y1] invariants were constructed: Definition 1 Let C be a tensor category. A functorial tensor automorphism of C is a natural isomorphism φ : 1C =⇒ 1C, which satisfies φA⊗B = φA⊗φB for all objects A and B, and φI = IdI , where I is the identity object for the tensor product. Observe that functorial tensor automorphisms for a subgroup of the abelian group of natural automorphisms of the identity functor on the category. The categories to which we will apply this notion are particularly nice, in that they satsify Definition 2 A k-linear abelian category C is semisimple if there exists a set J of objects such that 1. Every object is isomorphic to a finite direct sum of objects in J .