Relative torsion for representations in finite type Hilbert modules

TL;DR

This paper introduces the concept of relative torsion for representations in finite type Hilbert modules over closed manifolds, providing a new invariant that is always defined and relates to existing torsions when the pair is of determinant class.

Contribution

It defines the relative torsion invariant for a broad class of representations and computes its value, extending torsion theory beyond determinant class cases.

Findings

Relative torsion is always well-defined for the given representations.

When the pair is of determinant class, relative torsion equals the quotient of analytic and Reidemeister torsion.

The paper provides explicit calculations of the relative torsion.

Abstract

For a closed manifold equipped with a Riemannian metric, a triangulation, a representation of its fundamental group on an Hilbert module of finite type (over of finite von Neumann algebra), and a Hermitian structure on the flat bundle associated to the representation, one defines a numerical invariant, the relative torsion. The relative torsion is a positive real number and unlike the analytic torsion or the Reidemeister torsion, which are defined only when the pair manifold- representation is of determinant class, is always defined. When the pair is of determinant class the relative torsionis equal to the quotient of the analytic and the Reidemeister torsion.We calculate the relative torsion.