The Analytic Torsion on Manifolds with Boundary and Conical Singularities

Abstract

<p>The analytic torsion was introduced by D.B. Ray and I.M. Singer as an analytic counterpart to the combinatorial Reidemeister torsion. In this thesis we are concerned with analytic torsion of manifolds with boundary and conical singularities. Our work is comprised basically of three projects.<br /> In the first project we discuss a specific class of regular singular Sturm Liouville operators with matrix coefficients. Their zeta determinants were studied by K. Kirsten, P. Loya and J. Park on the basis of the Contour integral method, with general boundary conditions at the singularity and Dirichlet boundary conditions at the regular boundary.<br /> Our main result in the first project is the explicit verification that the Contour integral method indeed applies in the regular singular setup, and the generalization of the zeta determinant computations by Kirsten, Loya and Park to generalized Neumann boundary conditions at the regular boundary. Moreover we apply our results to Laplacians on a bounded generalized cone with relative boundary conditions.<br /> In the second project we derive a new formula for analytic torsion of a bounded generalized cone, generalizing the computational methods of M. Spreafico and using the symmetry in the de Rham complex, as established by M. Lesch. We evaluate our result in lower dimensions and further provide a separate computation of analytic torsion of a bounded generalized cone over S¹, since the standard cone over the sphere is simply a flat disc. <br /> Finally, in the third project we discuss the refined analytic torsion, introduced by M. Braverman and T. Kappeler as a canonical refinement of analytic torsion on closed manifolds. Unfortunately there seems to be no canonical way to extend their construction to compact manifolds with boundary.<br /> We propose a different refinement of analytic torsion, similar to Braverman and Kappeler, which does apply to compact manifolds with and without boundary. We establish a gluing formula for our construction, which in fact can also be viewed as a gluing law for the original definition of refined analytic torsion by Braverman and Kappeler.</p>