Reidemeister torsion and analytic torsion of discs

Details Reidemeister torsion and analytic torsion of discs Reidemeister torsion and analytic torsion of discs

2008-11-19

arxiv.org/abs/0811.3196v1

Mathematics

Differential Geometry

Scientific paper

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 \cite{RS}. We prove that the Reidemeister torsion coincides with a power 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-M\"{u}ller theorem. We use a formula proved by Br\"uning and Ma \cite{BM}, that predicts a new anomaly boundary term beside the known term proportional to the Euler characteristic of the boundary \cite{Luc}. 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 two 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 \cite{DF}, and we show that the result obtained using this formula is inconsistent with the direct calculation of the analytic torsion.

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