State-sum invariants of 3-manifolds associated to artinian semisimple tortile categories

AbstractThe method of Turaev and Viro is generalized to construct state-sum invariants of 3-manifolds using an artinian semisimple tortile category as initial data.In the first two sections of this paper we lay the topological and algebraic groundwork for the construction of a large class of (2+1)-dimensional TQFT's and their associated 3-manifold invariants. Then, following Turaev and Viro [21], we will first construct the 3-manifold invariants, and then handle the “relative case” to obtain TQFT's. In doing so, we isolate the analogues of the (classical or quantum) 6j-symbols in any artinian semisimple tortile category over an arbitrary field, in the correct normalization to repeat the construction of Turaev and Viro [21] with some modifications.Throughout, we assume all manifolds and homeomorphisms are piecewise linear and write compositions in diagrammatic order