[Paper Review] Relative torsion for representations in finite type Hilbert modules

Dan Burghelea, Leonid Friedlander+1 more|ArXiv.org|Nov 25, 1997

Advanced Operator Algebra Research8 references3 citations

TL;DR

This paper introduces a new numerical invariant called relative torsion for representations in finite type Hilbert modules over finite von Neumann algebras, defined on a closed manifold with a Riemannian metric, triangulation, and Hermitian structure on the associated flat bundle. Unlike analytic or Reidemeister torsion, it is always defined and equals the quotient of analytic and Reidemeister torsion when the pair is of determinant class, providing a unified framework for torsion invariants in non-determinant-class settings.

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.

Motivation & Objective

• To define a new invariant—relative torsion—that extends torsion theory beyond the determinant class condition.
• To address the limitation of analytic and Reidemeister torsion, which require the determinant class condition for definition.
• To unify analytic and Reidemeister torsion via a quotient that is always defined in the finite type Hilbert module setting.
• To provide a framework for torsion invariants in geometric and topological contexts involving von Neumann algebras and flat bundles.
• To generalize classical torsion invariants to representations in Hilbert modules of finite type.

Proposed method

• Define relative torsion as a positive real number associated with a closed manifold, Riemannian metric, triangulation, and a representation of the fundamental group on a finite type Hilbert module.
• Equip the flat bundle associated to the representation with a Hermitian structure to define the necessary geometric data.
• Construct the relative torsion using analytic and topological data, leveraging the structure of finite von Neumann algebras.
• Establish the invariance of the relative torsion under changes of metric and triangulation, ensuring its topological significance.
• Prove that when the pair (manifold, representation) is of determinant class, the relative torsion equals the quotient of analytic and Reidemeister torsion.
• Use techniques from global analysis and von Neumann algebra theory to ensure the invariant is well-defined and computable.

Experimental results

Research questions

• RQ1 Can a torsion invariant be defined in settings where the classical analytic and Reidemeister torsions fail due to the determinant class condition?
• RQ2 What is the relationship between the new relative torsion and the classical analytic and Reidemeister torsions when the determinant class condition holds?
• RQ3 How does the relative torsion behave under changes of metric and triangulation on the manifold?
• RQ4 Is the relative torsion invariant under isomorphisms of the flat bundle with Hermitian structure?
• RQ5 Can the relative torsion serve as a unifying invariant across different torsion theories in non-classical settings?

Key findings

• The relative torsion is always defined for any closed manifold equipped with a Riemannian metric, a triangulation, and a representation in a finite type Hilbert module.
• The relative torsion is a positive real number, independent of the choice of metric and triangulation, establishing its topological invariance.
• When the pair (manifold, representation) is of determinant class, the relative torsion equals the quotient of the analytic torsion and the Reidemeister torsion.
• The construction provides a natural extension of torsion invariants to cases where the classical invariants are not defined.
• The relative torsion unifies analytic and Reidemeister torsion in a single, consistent framework for finite type Hilbert modules.
• The method relies on the structure of finite von Neumann algebras and global analysis techniques to ensure well-definedness and invariance.