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Comparison of analytic and topological torsion on singular spaces

Final Report Summary - COMPTORSING (Comparison of analytic and topological torsion on singular spaces)

A Leitmotiv in global analysis are comparison results between analytic and topological invariants. One such comparison result is the the celebrated Cheeger-Müller theorem on the comparison of torsions for a smooth compact manifold proved in the 70ties by Cheeger and Müller independently. Comparison theorems between analytic and topological torsion on smooth manifolds have been an object of intensive study during the last 40 years in Mathematics. The most general result on comparison of torsion on a smooth compact manifold has been achieved by Bismut and Zhang combining local index techniques and the Witten deformation.

In recent years, the study of analytic torsion on singular spaces has gained interest. The idea of the project COMPTORSING is to approach the question of comparison of torsion for singular spaces with isolated cone-like singularities, using the so-called Witten deformation. The Witten deformation approach in the singular context is new and is based on the generalisation of the Witten deformation to singular spaces obtained by the fellow in a series of papers.

The Witten deformation generalised to the singular setting, consists of a deformation of the complex of L2-forms using a Morse function. In the context of the present question so-called radial and anti-radial Morse functions are particularly useful: near a singularity they only depend (quadratically!) on the distance from the singularity. The strong version of Witten's program can be generalised to singular spaces and anti-radial Morse functions: namely a comparison between a finite dimensional analytic complex (the complex of eigenforms of the Witten Laplacian to small eigenvalues) and a combinatorial complex. The later computes the intersection homology of the space, an important topological invariant of singular spaces.

One first successful line of research in the present project has been the systematic study of the combinatorial complex, associated to an anti-radial Morse function. This complex is generated by the smooth critical points of the Morse function in the interior and generators of the de Rham cohomology (in low degree) of the link manifolds of the singularities. As in smooth Morse theory one can prove a homotopy principle relating the complexes associated to two different choices of Morse function. The technical difficulty in this study is the generalisation of techniques developed by Laudenbach in the setting of smooth Morse theory to the singular context. The discussion of the ``homotopy principle'' naturally involves the study of a geometric complex on a space with non-isolated singularities.

A guiding principle of the Witten deformation is the localisation principle: eigenforms to small eigenvalues of the Witten Laplacian are concentrating near the critical points of the Morse function. The analytic study in this project has therefore been divided accordingly into a local and a global problem.

The local model situation has been studied with success applying local index techniques on the one hand. On the other hand the spectral theory of the model operator can be studied. The study of the Witten Laplacian using local index techniques allows to establish a formula relating the intersection homology of the cone with the ``Cheeger term'' and an integral over a certain Chern-Simons class. In case of a link manifold which is the boundary of a smooth manifold with boundary the formula can also be derived by comparing the index theorem of Cheeger for singular spaces with cone-like singularities and the index theorem for manifolds with boundary. In the general situation (in case of an arbitrary link manifold) the formula is new and of interest in itself. The study of the spectral theory of the model operator allows in particular to give an explicit formula for the analytic torsion of the model operator.

Important steps in the global study have been achieved as well, as the generalisation of the Berezin formalism to the singular setting (for all link manifolds). First steps towards the study of some of the topological aspects for more general singularities have been investigated as well.

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Current Address:
Ursula Ludwig
Mathematisches Institut
Universität Bonn
Endenicher Allee 60
53115 Bonn