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Assume temporarily that h is a flat metric on F . Let K be a smooth triangulation of Z . We can define the Reidemeistermetric ‖ ‖ λ(F ) on λ(F ). It is a basic result of Franz [13], Reidemeister [29], and de-Rham [30] (see also [25, §8]), that the metric ‖ ‖ λ(F ) does not depend on K . The metric ‖ ‖ λ(F ) on λ(F ) is then a topological invariant of F . If H (Z,F ) = 0, it is a positive number, now called the Reidemeister torsion (or Rtorsion).

Flat vector bundles and analytic torsion forms