Reidemeister Torsion and Analytic Torsion of Discs

T. de Melo; L. Hartmann; M. Spreafico

Bollettino dell'Unione Matematica Italiana (2009)

  • Volume: 2, Issue: 2, page 529-533
  • ISSN: 0392-4041

Abstract

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We study the Reidemeister torsion and the analytic torsion of the m-dimensional disc in the Euclidean m-dimensional space, using the base for the homology defined by Ray and Singer in [10]. We prove that the Reidemeister torsion coincides with the square root of the volume of the disc. We study the additional terms arising in the analytic torsion due to the boundary, using generalizations of the Cheeger-Müller theorem. We use a formula proved by Brüning and Ma [1], that predicts a new anomaly boundary term beside the known term proportional to the Euler characteristic of the boundary [6]. Some of our results extend to the case of the cone over a sphere, in particular we evaluate directly the analytic torsion for a cone over the circle and over the 2-sphere. We compare the results obtained in the low dimensional cases. We also consider a different formula for the boundary term given by Dai and Fang [4], and we show that the result obtained using this formula is inconsistent with the direct calculation of the analytic torsion.

How to cite

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de Melo, T., Hartmann, L., and Spreafico, M.. "Reidemeister Torsion and Analytic Torsion of Discs." Bollettino dell'Unione Matematica Italiana 2.2 (2009): 529-533. <http://eudml.org/doc/290559>.

@article{deMelo2009,
abstract = {We study the Reidemeister torsion and the analytic torsion of the m-dimensional disc in the Euclidean m-dimensional space, using the base for the homology defined by Ray and Singer in [10]. We prove that the Reidemeister torsion coincides with the square root of the volume of the disc. We study the additional terms arising in the analytic torsion due to the boundary, using generalizations of the Cheeger-Müller theorem. We use a formula proved by Brüning and Ma [1], that predicts a new anomaly boundary term beside the known term proportional to the Euler characteristic of the boundary [6]. Some of our results extend to the case of the cone over a sphere, in particular we evaluate directly the analytic torsion for a cone over the circle and over the 2-sphere. We compare the results obtained in the low dimensional cases. We also consider a different formula for the boundary term given by Dai and Fang [4], and we show that the result obtained using this formula is inconsistent with the direct calculation of the analytic torsion.},
author = {de Melo, T., Hartmann, L., Spreafico, M.},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {529-533},
publisher = {Unione Matematica Italiana},
title = {Reidemeister Torsion and Analytic Torsion of Discs},
url = {http://eudml.org/doc/290559},
volume = {2},
year = {2009},
}

TY - JOUR
AU - de Melo, T.
AU - Hartmann, L.
AU - Spreafico, M.
TI - Reidemeister Torsion and Analytic Torsion of Discs
JO - Bollettino dell'Unione Matematica Italiana
DA - 2009/6//
PB - Unione Matematica Italiana
VL - 2
IS - 2
SP - 529
EP - 533
AB - We study the Reidemeister torsion and the analytic torsion of the m-dimensional disc in the Euclidean m-dimensional space, using the base for the homology defined by Ray and Singer in [10]. We prove that the Reidemeister torsion coincides with the square root of the volume of the disc. We study the additional terms arising in the analytic torsion due to the boundary, using generalizations of the Cheeger-Müller theorem. We use a formula proved by Brüning and Ma [1], that predicts a new anomaly boundary term beside the known term proportional to the Euler characteristic of the boundary [6]. Some of our results extend to the case of the cone over a sphere, in particular we evaluate directly the analytic torsion for a cone over the circle and over the 2-sphere. We compare the results obtained in the low dimensional cases. We also consider a different formula for the boundary term given by Dai and Fang [4], and we show that the result obtained using this formula is inconsistent with the direct calculation of the analytic torsion.
LA - eng
UR - http://eudml.org/doc/290559
ER -

References

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  1. BRÜNING, J. - MA, XIAONAN, An anomaly formula for Ray-Singer metrics on manifolds with boundary, GAFA, 16 (2006) 767-873. Zbl1111.58024MR2255381DOI10.1007/s00039-006-0574-7
  2. CHEEGER, J., Analytic torsion and the heat equation, Ann. Math., 109 (1979) 259-322. Zbl0412.58026MR528965DOI10.2307/1971113
  3. CHEEGER, J., Spectral geometry of singular Riemannian spaces, J. Diff. Geom., 18 (1983) 575-657. Zbl0529.58034MR730920
  4. DAI, X. - FANG, H., Analytic torsion and R-torsion for manifolds with boundary, Asian J. Math., 4 (2000) 695-714. Zbl0996.58021MR1796700DOI10.4310/AJM.2000.v4.n3.a10
  5. HARTMANN, L. - DE MELO, T. - SPREAFICO, M., Reidemeister torsion and analytic torsion of discs, preprint (2008), arXiv:0811.3196v1. Zbl1181.58025MR2520992
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  8. MILNOR, J., Whitehead torsion, Bull. AMS, 72 (1966) 358-426. MR196736DOI10.1090/S0002-9904-1966-11484-2
  9. MÜLLER, W., Analytic torsion and R-torsion of Riemannian manifolds, Adv. Math., 28 (1978) 233-305. MR498252DOI10.1016/0001-8708(78)90116-0
  10. RAY, D. B. - SINGER, I. M., R-torsion and the Laplacian on Riemannian manifolds, Adv. Math., 7 (1971) 145-210. Zbl0239.58014MR295381DOI10.1016/0001-8708(71)90045-4
  11. RAY, D. B., Reidemeister torsion and the Laplacian on lens spaces, Adv. Math., 4 (1970) 109-126. Zbl0204.23804MR258062DOI10.1016/0001-8708(70)90018-6
  12. REIDEMEISTER, K., Homotopieringe und Linseräume, Hamburger Abhandl., 11 (1935) 102-109. MR3069647DOI10.1007/BF02940717
  13. SPREAFICO, M., Zeta function and regularized determinant on a disc and on a cone, J. Geo. Phys., 54 (2005) 355-371. MR2139088DOI10.1016/j.geomphys.2004.10.005
  14. SPREAFICO, M., Zeta invariants for Dirichlet series, Pacific. J. Math., 224 (2006) 180-199. Zbl1109.11055MR2231657DOI10.2140/pjm.2006.224.185
  15. SPREAFICO, M., Zeta invariants for double sequences of spectral type and a generalization of the Kronecker first limit formula, preprint (2006). Zbl1219.11134MR2250451DOI10.1017/S0308210500004777
  16. WENG, L. - YOU, Y., Analytic torsions of spheres, Int. J. Math., 7 (1996) 109-125. Zbl0854.58043MR1369907DOI10.1142/S0129167X96000074

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