Quantum Obstruction Theory

We use topological quantum eld theory to derive an invariant of a three-manifold with boundary. We then show how to use the structure of this invariant as an obstruction to embedding one three-manifold into another. 1. Introduction In the mid 1980's the Jones polynomial of a link was introduced 5], 6]. It was deened as the normalized trace of an element of a braid group, corresponding to the link, in a certain representation. Although a topological invariant, the extrinsic nature of its computation made the relationship between topological conngurations lying in the complement of the knot and the value of the Jones polynomial obscure. In order to elucidate this connection, Witten 20] introduced Topological Quantum Field Theory. TQFT is a cut and paste technique which allows for localized computation of the Jones polynomial. Following his discovery various approaches to topological quantum eld theory and others). In the work of Witten, it is obvious that the vector spaces of topologi-cal quantum eld theory are quantizations of the space of connections on the manifold. Initial considerations of TQFT focused on the formal rules of combination of vector spaces and vectors. As more intrinsic developments of TQFT appeared there was a shift towards the use of the Kauuman bracket skein module. The Kauuman bracket skein module of a 3-manifold can be thought of as a quantization of the SL 2 (C){ characters of the manifold 2]. This approach to TQFT recovers some of the initial feel of Witten's work. Although the study of 3-manifold invariants is substantial in itself, few applications to classical 3-manifold topology have been found. In