John M. Bryden - 3-manifold invariants associated to topological quantum field theories

Let M be a closed orientable 3-manifold obtained from surgery on a framed link. Given a primitive nth (resp. 2n-th) root of unity, q, for an odd (resp. even) positive integer n, there are a series of invariants, Z_n(M,q), defined by Murakami, Ohtsuki and Okada which are invariant up to homotopy type. The definition of these invariants is in terms of Gauss sums. In fact these invariants have now been generalized by Deloup to give a new abelian quantum invariant which can be expressed as a product of Gauss sums. These invariants can be determined from the linking matrix and the cohomolgy algebra of the manifold. When M is a Seifert manifold joint work with C. Hayat-Legrand, H. Zieschang and P. Zvengrowski can be used to complete the computation of these invariants in a simple way. This framework can also be used to calculate the Dijkgraaf-Witten invariants for a class of topological quantum field theories, as long as M is a Seifert manifold.

Although there has been some progress made in understanding the combinatorial nature of these and other quantum invariants, their geometric nature and their relationship to the fundamental group and to cohomology is not understood. An initial attempt to understand these invariants in the context of algebraic topology is to interpret the surgery in the cohomology algebras of Seifert manifolds.