Comparison of the refined analytic and the Burghelea-Haller torsions

Braverman, Maxim; Kappeler, Thomas

Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2361-2387.

La torsion analytique raffinée, associée à un fibré vectoriel plat sur une variété fermée et orientée de dimension impaire, définit d’une manière canonique une forme quadratique $\tau$ sur le déterminant de la cohomologie. La torsion introduite par Burghelea et Haller et la forme quadratique $\tau$ sont des concepts raffinés de la torsion analytique de Ray-Singer. On démontre que dans le cas où la torsion de Burghelea-Haller est définie, elle est identique à $\pm \tau$. Comme application, on obtient des résultats nouveaux pour la torsion de Burghelea-Haller. En particulier, on démontre une version faible de la conjecture de Burghelea-Haller concernant leur torsion et le carré de la torsion combinatoire de Farber-Turaev.

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.

Classification : 58J52, 58J28, 57R20
Keywords: Determinant line, analytic torsion, Ray-Singer torsion, eta-invariant, Turaev torsion and Farber-Turaev torsion
Mot clés : déterminant, torsion analytique, torsion de Ray-Singer, invariant eta, torsion de Turaev et de Farber-Turaev

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