  Let (M,g) be an odd-dimensional incomplete compact Riemannian singular space
with a simple edge singularity. We study the analytic torsion on M, and in
particular consider how it depends on the metric g. If g is an admissible edge
metric, we prove that the torsion zeta function is holomorphic near s = 0,
hence the torsion is well-defined, but possibly depends on g. In general
dimensions, we prove that the analytic torsion depends only on the asymptotic
structure of g near the singular stratum of M; when the dimension of the edge
is odd, we prove that the analytic torsion is independent of the choice of
admissible edge metric. The main tool is the construction, via the methodology
of geometric microlocal analysis, of the heat kernel for the Friedrichs
extension of the Hodge Laplacian in all degrees. In this way we obtain detailed
asymptotics of this heat kernel and its trace.
