Comparison of the refined analytic and the Burghelea-Haller torsions
Annales de l'Institut Fourier, Volume 57 (2007) no. 7, p. 2361-2387

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.

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.

DOI : https://doi.org/10.5802/aif.2336
Classification:  58J52,  58J28,  57R20
Keywords: Determinant line, analytic torsion, Ray-Singer torsion, eta-invariant, Turaev torsion and Farber-Turaev torsion
@article{AIF_2007__57_7_2361_0,
     author = {Braverman, Maxim and Kappeler, Thomas},
     title = {Comparison of the refined analytic and the Burghelea-Haller torsions},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {7},
     year = {2007},
     pages = {2361-2387},
     doi = {10.5802/aif.2336},
     mrnumber = {2394545},
     zbl = {1147.58033},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2007__57_7_2361_0}
}
Braverman, Maxim; Kappeler, Thomas. Comparison of the refined analytic and the Burghelea-Haller torsions. Annales de l'Institut Fourier, Volume 57 (2007) no. 7, pp. 2361-2387. doi : 10.5802/aif.2336. http://www.numdam.org/item/AIF_2007__57_7_2361_0/

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