Comparison of the refined analytic and the Burghelea-Haller torsions
[Comparaison de la torsion raffinée et la torsion de Burghelea-Haller]
Braverman, Maxim1 ; Kappeler, Thomas2
1 Northeastern University Department of Mathematics Northeastern University Boston, MA 02115 (USA)
2 Universität Zürich Institut für Mathematik Winterthurerstrasse 190 8057 Zürich (Switzerland)
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 τ sur le déterminant de la cohomologie. La torsion introduite par Burghelea et Haller et la forme quadratique τ 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 ±τ. 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 τ on the determinant line of the cohomology. Both τ 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 ±τ. 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.

DOI : 10.5802/aif.2336
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
Braverman, Maxim 1 ; Kappeler, Thomas 2

1 Northeastern University Department of Mathematics Northeastern University Boston, MA 02115 (USA)
2 Universität Zürich Institut für Mathematik Winterthurerstrasse 190 8057 Zürich (Switzerland) 
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Braverman, Maxim; Kappeler, Thomas. Comparison of the refined analytic and the Burghelea-Haller torsions. Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2361-2387. doi : 10.5802/aif.2336. https://www.numdam.org/articles/10.5802/aif.2336/

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