Extra structures on three-dimensional cobordisms

Abstract

A Topological Quantum Field Theory (TQFT) is a functor from a cobordism category to the category of vector spaces, satisfying certain properties. An important property is that the vector spaces should be finite dimensional. For the WRT TQFT, the relevant 2 + 1-cobordism category is built from manifolds which are equipped with an extra structure such as a p1-structure, or an extended manifold structure. In chapter 1, we perform the universal construction of [3] on a cobordism category without this extra structure and show that the resulting quantization functor assigns an infinite dimensional vector space to the torus. In chapter 2, we enhance the extended manifold structure through introducing oriented lagrangians. We apply a machinery introduced by Guillemin and Sternberg in [7] to transport oriented lagrangians. Using Lion and Vergne’s s map in [12, p66], we defined a modulo 4 invariant for cobordisms equipped with such an enhanced structure. This invariant can be viewed as a generalization of Gilmer and Masbaum’s n_x0015_(f) in [6, Theorem 6.6], which is defined on the extended mapping class group. The techniques used here might be useful in finding a index 4 subcategory