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.
@article{BUMI_2009_9_2_2_529_0,
author = {T. de Melo and L. Hartmann and M. Spreafico},
title = {Reidemeister Torsion and Analytic Torsion of Discs},
journal = {Bollettino dell'Unione Matematica Italiana},
volume = {2},
year = {2009},
pages = {529-533},
zbl = {1181.58025},
mrnumber = {2537286},
language = {en},
url = {http://dml.mathdoc.fr/item/BUMI_2009_9_2_2_529_0}
}
de Melo, T.; Hartmann, L.; Spreafico, M. Reidemeister Torsion and Analytic Torsion of Discs. Bollettino dell'Unione Matematica Italiana, Tome 2 (2009) pp. 529-533. http://gdmltest.u-ga.fr/item/BUMI_2009_9_2_2_529_0/
[1] - , An anomaly formula for Ray-Singer metrics on manifolds with boundary, GAFA, 16 (2006) 767-873. | MR 2255381 | Zbl 1111.58024
[2] , Analytic torsion and the heat equation, Ann. Math., 109 (1979) 259-322. | MR 528965 | Zbl 0412.58026
[3] , Spectral geometry of singular Riemannian spaces, J. Diff. Geom., 18 (1983) 575-657. | MR 730920 | Zbl 0529.58034
[4] - , Analytic torsion and R-torsion for manifolds with boundary, Asian J. Math., 4 (2000) 695-714. | MR 1796700 | Zbl 0996.58021
[5] - - , Reidemeister torsion and analytic torsion of discs, preprint (2008), arXiv:0811.3196v1. | MR 2520992 | Zbl 1181.58025
[6] , Analytic and topological torsion for manifolds with boundary and symmetry, J. Differential Geom., 37 (1993) 263-322. | MR 1205447
[7] - , Reidemeister torsion and analytic torsions of spheres, preprint 2008. | MR 2520992 | Zbl 1201.57015
[8] , Whitehead torsion, Bull. AMS, 72 (1966) 358-426. | MR 196736
[9] , Analytic torsion and R-torsion of Riemannian manifolds, Adv. Math., 28 (1978) 233-305. | MR 498252
[10] - , R-torsion and the Laplacian on Riemannian manifolds, Adv. Math., 7 (1971) 145-210. | MR 295381 | Zbl 0239.58014
[11] , Reidemeister torsion and the Laplacian on lens spaces, Adv. Math., 4 (1970) 109-126. | MR 258062 | Zbl 0204.23804
[12] , Homotopieringe und Linseräume, Hamburger Abhandl., 11 (1935) 102-109. | MR 3069647
[13] , Zeta function and regularized determinant on a disc and on a cone, J. Geo. Phys., 54 (2005) 355-371. | MR 2139088
[14] , Zeta invariants for Dirichlet series, Pacific. J. Math., 224 (2006) 180-199. | MR 2231657 | Zbl 1109.11055
[15] , Zeta invariants for double sequences of spectral type and a generalization of the Kronecker first limit formula, preprint (2006). | MR 2250451 | Zbl 1219.11134
[16] - , Analytic torsions of spheres, Int. J. Math., 7 (1996) 109-125. | MR 1369907 | Zbl 0854.58043