Invariants of Piecewise - Linear 3 - Manifolds

The purpose of this paper is to present an algebraic framework for constructing invariants of closed oriented 3-manifolds. The construction is in the spirit of topological field theory and the invariant is calculated from a triangulation of the 3-manifold. The data for the construction of the invariant is a tensor category with a condition on the duals, which we have called a spherical category. The definition of a spherical category and a coherence theorem needed in this paper are given in (Barrett and Westbury [1993]). There are two classes of examples of spherical categories discussed in this paper. The first examples are given by the quantised enveloping algebra of a semisimple Lie algebra, and the second are given by an involutive Hopf algebra. In the first case, the invariant for sl2 defined in this paper is the Turaev-Viro invariant (Turaev and Viro [1992]). This invariant is known to distinguish lens spaces of the same homotopy type which already shows that the invariants in this paper are not trivial. The problem of generalising the Turaev-Viro invariant to other quantised enveloping algebras has also been considered by (Durhuus, Jakobsen and Nest [1993]) and (Yetter [1993]). A noteworthy feature of our construction is that it does not require a braiding; the notion of a spherical category is more general than the notion of a ribbon category. A simple example of this is the category of representations of the convolution algebra of a non-abelian finite group. This is a spherical category but does not admit a braiding because the representation ring of the algebra is not commutative.