[Paper Review] Poincaré - Reidemeister metric, Euler structures, and torsion

Michael Färber, Vladimir Turaev|ArXiv.org|Mar 27, 1998

Homotopy and Cohomology in Algebraic Topology10 references17 citations

TL;DR

This paper introduces the Poincaré-Reidemeister (PR) scalar product on the determinant line of cohomology for flat vector bundles over odd-dimensional, closed, orientable manifolds, refining the PR-norm by incorporating sign or phase information. It establishes a formula expressing the PR-scalar product in terms of torsions of Euler structures, showing its sign is determined by Stiefel-Whitney classes and the manifold's semi-characteristic, and applies this to compute Ray-Singer analytic torsion and twisted semi-characteristics modulo 2.

ABSTRACT

In this paper we define a Poincaré-Reidemeister scalar product on the determinant line of the cohomology of any flat vector bundle over a closed orientable odd-dimensional manifold. It is a combinatorial "torsion-type" invariant which refines the PR-metric, introduced earlier by the first author, and contains an additional sign or phase information. We compute the PR-scalar product in terms of the torsions of Euler structures, introduced earlier by the second author. We show that the sign of our PR-scalar product is determined by the Stiefel-Whitney classes and the semi-characteristic of the manifold. As an application, we compute the Ray-Singer analytic torsion via the torsions of Euler structures. Another application: a computation of the twisted semi-characteristic in terms of the Stiefel-Whitney classes.

Motivation & Objective

• To define a refined Poincaré-Reidemeister scalar product on the determinant line of cohomology that includes sign or phase information beyond the norm.
• To establish a precise formula expressing the PR-scalar product in terms of torsions of Euler structures on odd-dimensional manifolds.
• To compute the Ray-Singer analytic torsion using torsions of Euler structures.
• To derive a formula for the twisted semi-characteristic modulo 2 in terms of Stiefel-Whitney classes of the manifold and flat bundle.
• To develop algebraic tools for handling sign anomalies in determinant line formalism, particularly in duality and duality maps.

Proposed method

• Define the Poincaré-Reidemeister scalar product on the determinant line of cohomology using a combination of Reidemeister torsion and Poincaré duality.
• Introduce and systematically treat torsions of Euler structures as elements in the determinant line of homology, with explicit dependence on a characteristic homology class $ c(\xi) \in H_1(X) $.
• Use duality maps and isomorphisms between determinant lines of homology and cohomology to relate torsions of dual bundles and establish sign consistency.
• Apply the fusion isomorphism and sign tracking via the $ s(V_p) $ invariant to compute the degree of the composition of duality and bundle duality maps.
• Leverage the main result of Farber [Fa] that the PR-norm coincides with the Ray-Singer norm to compute analytic torsion via Euler structure torsions.
• Use the duality between cohomological and homological formulations to reformulate the main theorems in cohomological terms.

Experimental results

Research questions

• RQ1 How can the Poincaré-Reidemeister scalar product be defined to include sign or phase information beyond the norm?
• RQ2 What is the precise relationship between the PR-scalar product and the torsions of Euler structures on odd-dimensional manifolds?
• RQ3 How does the sign of the PR-scalar product depend on the Stiefel-Whitney classes and the semi-characteristic of the manifold?
• RQ4 Can the Ray-Singer analytic torsion be computed using torsions of Euler structures?
• RQ5 What is the formula for the twisted semi-characteristic modulo 2 in terms of Stiefel-Whitney classes?

Key findings

• The sign of the Poincaré-Reidemeister scalar product is determined by the Stiefel-Whitney classes $ w_1(F) $, $ w_{m-1}(X) $, and the semi-characteristic of the manifold $ X $.
• For even-dimensional flat bundles, the PR-scalar product applied to the torsion of an Euler structure $ \xi $ is expressed via the determinant of the characteristic class $ c(\xi) \in H_1(X) $.
• For odd-dimensional bundles, the PR-scalar product depends on both the Euler structure $ \xi $ and a choice of homology orientation of $ X $.
• The Ray-Singer analytic torsion is computed as a signed sum over torsions of Euler structures, with signs determined by the duality and bundle isomorphism data.
• The twisted semi-characteristic $ s\chi_F(X) \mod 2 $ equals $ \langle w_1(F) \cup w_{m-1}(X), [X] \rangle + s\chi(X) \cdot \dim F \mod 2 $ when $ m \equiv 1 \mod 4 $.
• The proof establishes that the degree of the composition $ \phi_* \circ D $ on the determinant line is $ (-1)^{s\chi_F(X)} $, which matches the sign from the Stiefel-Whitney class formula.