9626818 Kania-Bartoszynska This project belongs in the general area of low-dimensional topology. More specifically, it deals with applications of topological quantum field theory to classical three-manifold topology. The investigator will use quantum invariants to construct obstructions to the embedding of three-manifolds in the three-sphere and to construct invariants for the detection of lens spaces and Seifert fiber spaces. Additionally, quantum invariants are to be used to formulate criteria for the existence of symmetries in three-manifolds. Low-dimensional topology is mainly concerned with classifying spaces of dimensions three and four, where two spaces are treated as equivalent if one space can be deformed onto the other without tearing - this is called a homeomorphic transformation. Such a classification, known as the topological classification, has been essentially carried out in dimensions other than three and four. It turns out that these two dimensions provide various further technical difficulties, some of which are topics of much current interest as they interface with deep ideas of modern physics.
