Torsion of manifolds

This is a proposal for an FWF project that should take place at the Department of Mathematics at the University of Vienna. The aim is to study torsion of manifolds from different points of view. There is the well understood concept of combinatorial torsion associated to a manifold and a flat complex vector bundle which, for instance, can be defined with the help of a MorseSmale vector field. On the other hand we have the RaySinger torsion, defined with the help of analysis. According to a theorem of Cheeger, Müller and BismutZhang the RaySinger torsion essentially computes the absolute value of the combinatorial torsion. Using non-selfadjoint Laplacians, we have recently introduced a complex valued analytic torsion, which computes the combinatorial torsion, including its phase. We plan to investigate this complex valued analytic torsion further, establish the corresponding CheegerMüller type theorem in full generality and provide a generalization to bordisms. Another approach to torsion we are interested in is the one via gradients of circle valued Morse functions initiated by Freed. Assuming the vector field satisfies an exponential growth condition it permits to define a dynamical torsion. This torsion has an additional term given by the zeta function associated to the closed trajectories. In simple situations we could show that this dynamical torsion coincides with the combinatorial torsion. We aim at establishing this relation in general. The proposed projected would provide funds to employ two PhD students working on related subjects.

The aim of this research project was to investigate a particular link between the geometry and topology of manifolds. Manifolds are higher dimensional generalizations of curves and surfaces. To get an intuitive picture, think of the surface of a donut smoothly embedded in three dimensional space. Most importantly, manifolds locally look like familiar Euclidean space. Globally, however, their shape can be much more complicated. On manifolds one can do (coordinate free) calculus, whence the concept is fundamental to (natural) sciences. The aforementioned connection between geometry and topology is expressed as an equality of two, seemingly unrelated numbers; one defined using geometry, the other one defined in terms of topology. Many similar relations have been discovered and explored, the Euler characteristics provides a prototypical example: Consider a closed (finite and without boundary) surface which is smoothly embedded in three dimensional space. At each point on this surface we have a curvature measuring to what extent the surface deviates from being flat at this point. While this curvature function depends very much on the way the surface is embedded, the average curvature turns out to coincide with the Euler characterstics, F-E+V, where F denotes the number of faces, E the number of edges and V the number of vertices of a triangulation (tiling) of the surface. The stunning fact that these two numbers coincide, is known as the Gauss-Bonnet-Chern theorem. In particular, it provides an explenation of the following two facts: The average curvature does not depend on the way the surface is embedded, and the number F-E+V does not depend on the triangulation. The invariant we have been investigating is called torsion and actually related to the Euler characteristics. The Reidemeister torsion is a complex number defined in terms of the combinatorics of a triangulation (adjacency of tiles). The Ray-Singer torsion, on the other hand, is a positiv real number defined in terms of the spectrum of the Laplace-deRham operator, a differential operator closely related to geometry. Its spectrum consists of a discrete infinite number of frequencies, very much like the frequencies one can hear when hitting a drum. A celebrated result due to Cheeger, Müller and Bismut-Zhang asserts that the Ray-Singer torsion coincides with the absolut value of the Reidemeister torsion, expressing a profound link between topology and geometry. Nothing said so far is new. What we actually did, was to put forward and investigate a complex valued analog of the Ray-Singer torsion, expressing the full Reidemeister torsion (its absolut value and phase) in terms of spectral geometry. Our results can be understood as a tightening of a previously known link between geometry and topology.