Comparison of the refined analytic and the Burghelea-Haller torsions

Details Comparison of the refined analytic and the Burghelea-Haller torsions Comparison of the refined analytic and the Burghelea-Haller torsions

2006-06-16

arxiv.org/abs/math/0606398v2

Mathematics

Differential Geometry

To appear in Annales de l'institut Fourier. Compared to the first version many statements are refined and improved

Scientific paper

The refined analytic torsion associated to a flat vector bundle over a closed odd-dimensional manifold canonically defines a quadratic form $\tau$ on the determinant line of the cohomology. Both $\tau$ and the Burghelea-Haller torsion are refinements of the Ray-Singer torsion. We show that whenever the Burghelea-Haller torsion is defined it is equal to $\pm\tau$. As an application we obtain new results about the Burghelea-Haller torsion. In particular, we prove a weak version of the Burghelea-Haller conjecture relating their torsion with the square of the Farber-Turaev combinatorial torsion.

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