Turaev Torsion of 3-Manifolds with Boundary

Abstract

We study the Turaev torsion of 3-manifolds with boundary; specifically how certain ``leading order'' terms of the torsion are related to cohomology operations. Chapter 1 consists mainly of definitions and known results, providing some proofs of known results when the author hopes to present a new perspective. Chapter 2 deals with generalizations of some results of Turaev. Turaev's results relate leading order terms of the Turaev torsion of closed, oriented, connected 3-manifolds to certain ``determinants'' derived from cohomology operations such as the alternate trilinear form on the first cohomology group given by cup product. These determinants unfortunately do not generalize directly to compact, connected, oriented 3-manifolds with nonempty boundary, because one must incorporate the cohomology of the manifold relative to its boundary. We define the new determinants that will be needed, and show that with these determinants enjoy a similar relationship to the one given by Turaev between torsion and the known determinants. These definitions and results are given for integral cohomology, cohomology with mod-r coefficients for certain integers r, and for integral Massey products. Chapter 3 shows how to use the results of Chapter 2 to derive Turaev's results for integral cohomology, by studying how the determinant defined in Chapter 2 changes when gluing solid tori along boundary components, and also how this determinant is related to Turaev's determinant when one glues enough solid tori along the boundary to obtain a closed 3-manifold. One can then use known gluing formulae for torsion to derive Turaev's results relating torsion and cohomology of closed 3-manifolds to the results in Chapter 2.