[Paper Review] The Intersection R-Torsion for Finite Cone

Xianzhe Dai, Xiaoling Huang|arXiv (Cornell University)|Oct 22, 2014

Geometric and Algebraic Topology29 references3 citations

TL;DR

This paper establishes a formula for the intersection R-torsion of a finite cone in terms of determinants of combinatorial Laplacians on its cross-section, and introduces an analogous analytic invariant using Hodge Laplacians on even-dimensional manifolds. The key contribution is a precise expression linking topological torsion to spectral data, with applications to Mayer-Vietoris sequences and metric variation under conformal changes.

ABSTRACT

We prove a formula for the intersection R-torsion of a finite cone and use it to introduce a family of spectral invariants which is closely related to Cheeger's half torsion.

Motivation & Objective

• To compute the intersection R-torsion of a finite cone using combinatorial Laplacians on its cross-section.
• To define an analytic analog of the intersection R-torsion using Hodge Laplacians on even-dimensional manifolds.
• To study the variation of this analytic invariant under metric deformations, particularly conformal changes.
• To relate the R-torsion of the Mayer-Vietoris sequence in intersection cohomology to the R-torsion of the relative cohomology of a pair (M,Y).
• To establish conditions under which the R-torsion of the Mayer-Vietoris sequence becomes trivial or computable via truncated exact sequences.

Proposed method

• Expresses the intersection R-torsion $ I\tau^{\bar{p}}(X) $ of a finite cone $ X = C(Y) $ as a sum of logarithms of determinants of combinatorial Laplacians $ \Delta_p^{(c)} $ on the cross-section $ Y $, with coefficients depending on the perversity $ \bar{p} $.
• Introduces an analytic invariant $ \ln T_p(Y,\rho) $ by replacing combinatorial Laplacians with Hodge Laplacians $ \Delta_k $ on $ k $-forms with coefficients in a flat vector bundle $ E_\rho $.
• Derives the variation of $ \ln T_p(Y,\rho) $ under a one-parameter family of metrics $ g(u) $, involving traces of projections onto cohomology and constant terms in heat kernel expansions.
• Applies the Mayer-Vietoris sequence in intersection cohomology to relate the torsion of the cone to the relative cohomology of a pair $ (M,Y) $, under the Witt condition $ H^{m/2}(Y) = 0 $.
• Uses Hodge decomposition of heat kernels $ E_k = E_k^{ex} + E_k^{ce} + E_k^h $ to isolate the singular part in the variation formula.
• Employs split short exact sequences and preferred bases to compute R-torsions via transition matrices between bases, showing triviality under isometric conditions.

Experimental results

Research questions

• RQ1 How can the intersection R-torsion of a finite cone be expressed in terms of spectral data on its cross-section?
• RQ2 What is the analytic counterpart of the intersection R-torsion for even-dimensional manifolds, and how does it behave under metric changes?
• RQ3 Under what conditions does the R-torsion of the Mayer-Vietoris sequence in intersection cohomology reduce to the R-torsion of a relative cohomology sequence?
• RQ4 How does the variation of the analytic torsion invariant $ \ln T_p(Y,\rho) $ depend on the metric deformation, especially in the conformal case?
• RQ5 When is the analytic invariant $ \ln T_p(Y,\rho) $ locally dependent on the metric, and what role does the Witt condition play?

Key findings

• The intersection R-torsion of a finite cone is given by $ \ln I\tau^{\bar{p}}(X) = \sum_{p=0}^{n-p_n-1}(-1)^{p+1}p\ln\det\Delta_p^{(c)} + (n-p_n)\sum_{p=p_n}^{n-1}(-1)^{p+1}\ln\det\Delta_p^{(c)} $, explicitly linking it to combinatorial Laplacian determinants.
• The analytic invariant $ \ln T_p(Y,\rho) $ is defined as a combination of logarithmic determinants of Hodge Laplacians, generalizing the usual analytic torsion for $ p=0 $.
• The variation of $ \ln T_p(Y,\rho) $ under metric change involves traces of projection operators onto cohomology and constant terms in heat kernel expansions, with a clean local formula for $ p = m/2 $.
• For $ p = m/2 $, the variation becomes local: $ \frac{d}{du}\ln T_{m/2}(Y,\rho) = \frac{1}{2}\sum_{k=0}^{m/2-1}(-1)^{k+1}\operatorname{Tr}(P_{H^k}\alpha) + \frac{1}{2}\sum_{k=0}^{m/2-1}(-1)^{k+1}\operatorname{LIM}_{t\to 0}\operatorname{Tr}(e^{-t\Delta_k}\alpha) $.
• Under the Witt condition $ H^{m/2}(Y) = 0 $, the R-torsion of the Mayer-Vietoris sequence in intersection cohomology equals the R-torsion of the truncated relative cohomology sequence of $ (M,Y) $.
• The R-torsion of a split short exact sequence is trivial when the projection and inclusion maps preserve the preferred bases up to isometry, as shown via transition matrices with unit diagonal entries.