Reidemeister torsion and analytic torsion of discs
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 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\"uller theorem. We use a formula proved by Br\"uning and Ma BM, that predicts a new anomaly boundary term beside the known term proportional to the Euler characteristic of the boundary 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 DF, and we show that the result obtained using this formula is inconsistent with the direct calculation of the analytic torsion.