The refined analytic torsion associated to a flat vector bundle over a closed odd-dimensional manifold canonically defines a quadratic form on the determinant line of the cohomology. Both 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 . 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 sur le déterminant de la cohomologie. La torsion introduite par Burghelea et Haller et la forme quadratique 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 . 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.
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
@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}, pages = {2361--2387}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {57}, number = {7}, year = {2007}, doi = {10.5802/aif.2336}, mrnumber = {2394545}, zbl = {1147.58033}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2336/} }
TY - JOUR AU - Braverman, Maxim AU - Kappeler, Thomas TI - Comparison of the refined analytic and the Burghelea-Haller torsions JO - Annales de l'Institut Fourier PY - 2007 SP - 2361 EP - 2387 VL - 57 IS - 7 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2336/ DO - 10.5802/aif.2336 LA - en ID - AIF_2007__57_7_2361_0 ER -
%0 Journal Article %A Braverman, Maxim %A Kappeler, Thomas %T Comparison of the refined analytic and the Burghelea-Haller torsions %J Annales de l'Institut Fourier %D 2007 %P 2361-2387 %V 57 %N 7 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2336/ %R 10.5802/aif.2336 %G en %F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.2336/
[1] Heat kernels and Dirac operators, Grundlehren Text Editions, Springer-Verlag, Berlin, 2004 (Corrected reprint of the 1992 original) | MR | Zbl
[2] An extension of a theorem by Cheeger and Müller, Astérisque (1992) no. 205, pp. 235 (With an appendix by François Laudenbach) | MR | Zbl
[3] A canonical quadratic form on the determinant line of a flat vector bundle (arXiv:math.DG/0710.1232)
[4] Refined Analytic Torsion (arXiv:math.DG/0505537, To appear in J. of Differential Geometry)
[5] A refinement of the Ray-Singer torsion, C. R. Math. Acad. Sci. Paris, Volume 341 (2005) no. 8, pp. 497-502 | MR | Zbl
[6] Ray-Singer type theorem for the refined analytic torsion, J. Funct. Anal., Volume 243 (2007) no. 1, pp. 232-256 | DOI | MR | Zbl
[7] Refined analytic torsion as an element of the determinant line, Geom. Topol., Volume 11 (2007), pp. 139-213 | DOI | MR | Zbl
[8] Removing metric anomalies from Ray-Singer torsion, Lett. Math. Phys., Volume 47 (1999) no. 2, pp. 149-158 | DOI | MR | Zbl
[9] Asymptotic expansion of the Witten deformation of the analytic torsion, J. Funct. Anal., Volume 137 (1996) no. 2, pp. 320-363 | DOI | MR | Zbl
[10] Torsion, as a function on the space of representations (arXiv:math.DG/0507587)
[11] Euler structures, the variety of representations and the Milnor-Turaev torsion, Geom. Topol., Volume 10 (2006), p. 1185-1238 (electronic) | DOI | MR | Zbl
[12] Complex-valued Ray-Singer torsion, J. Funct. Anal., Volume 248 (2007) no. 1, pp. 27-78 | DOI | MR | Zbl
[13] Analytic torsion and the heat equation, Ann. of Math. (2), Volume 109 (1979) no. 2, pp. 259-322 | DOI | MR | Zbl
[14] Absolute torsion and eta-invariant, Math. Z., Volume 234 (2000) no. 2, pp. 339-349 | DOI | MR | Zbl
[15] Absolute torsion, Tel Aviv Topology Conference: Rothenberg Festschrift (1998) (Contemp. Math.), Volume 231, Amer. Math. Soc., Providence, RI, 1999, pp. 73-85 | MR | Zbl
[16] Poincaré-Reidemeister metric, Euler structures, and torsion, J. Reine Angew. Math., Volume 520 (2000), pp. 195-225 | DOI | MR | Zbl
[17] The eta invariant and secondary characteristic classes of locally flat bundles, Algebraic and differential topology-global differential geometry (Teubner-Texte Math.), Volume 70, Teubner, Leipzig, 1984, pp. 49-87 | MR | Zbl
[18] Refined analytic torsion: comparison theorems and examples (arXiv:math.DG/0602231, To appear in Illinois J. Math.)
[19] -invariant and flat vector bundles II (arXiv:math.DG/0604357) | MR
[20] Superconnections, Thom classes, and equivariant differential forms, Topology, Volume 25 (1986) no. 1, pp. 85-110 | DOI | MR | Zbl
[21] Analytic torsion and -torsion of Riemannian manifolds, Adv. in Math., Volume 28 (1978) no. 3, pp. 233-305 | DOI | MR | Zbl
[22] Analytic torsion and -torsion for unimodular representations, J. Amer. Math. Soc., Volume 6 (1993) no. 3, pp. 721-753 | DOI | MR | Zbl
[23] -torsion and the Laplacian on Riemannian manifolds, Advances in Math., Volume 7 (1971), pp. 145-210 | DOI | MR | Zbl
[24] Reidemeister torsion in knot theory, Russian Math. Survey, Volume 41 (1986), pp. 119-182 | DOI | MR | Zbl
[25] Euler structures, nonsingular vector fields, and Reidemeister-type torsions, Math. USSR Izvestia, Volume 34 (1990), pp. 627-662 | DOI | MR | Zbl
[26] Introduction to combinatorial torsions, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2001 (Notes taken by Felix Schlenk) | MR | Zbl
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