[Paper Review] Linear relations, monodromy and Jordan cells of a circle valued map

Dan Burghelea|arXiv (Cornell University)|Jan 11, 2015

Topological and Geometric Data Analysis12 references3 citations

TL;DR

This paper introduces a new algorithm for computing the Jordan cells of a circle-valued map using linear relations, providing a geometric proof of monodromy's homotopy invariance without relying on infinite cyclic covers or graph representations. The method enables efficient computation of topological invariants such as Novikov Betti numbers and Alexander polynomials via simplicial maps on finite complexes.

ABSTRACT

In this paper we consider the definition of " monodromy of an angle valued map" based on linear relations as proposed in Burghelea-Haller (3). This definition provides an alternative treatment of monodromy and computationally an alternative calculation of the "Jordan cells", topological persistence invariants of a circle valued maps introduced in Burghelea-Day (2). We give a new geometric proof that the monodromy is actually a homotopy invariant of a pair consisting of a compact ANR and an integral degree one cohomology class without any reference to the infinite cyclic cover associated to cohomology class as in (3), or to the graph representation associated an angle valued map defining the cohomology class as in (2). Most important, we describe an algorithm to calculate the monodromy for a simplicial angle valued map defined on a finite simplicial complex, providing a new algorithm for the calculation of the Jordan cells of the map, shorter than the one proposed in (2). We indicate the computational usefulness of "Jordan cells", and in particular of the proposed algorithm, for the calculation of other basic topological invariants of the pair.

Motivation & Objective

• To provide a new geometric proof that monodromy is a homotopy invariant of the pair $(X, ilde{ heta})$ without using infinite cyclic covers or graph representations.
• To develop an algorithm for computing the Jordan cells of a simplicial angle-valued map on a finite simplicial complex, offering a shorter computation path than prior methods.
• To demonstrate the computational utility of Jordan cells in calculating fundamental topological invariants such as Novikov Betti numbers, $L^2$-Betti numbers, and Alexander polynomials.
• To establish that the monodromy and Jordan cells depend only on the pair $(X, ilde{ heta})$, not on the specific representative map $f: X \to \mathbb{S}^1$, via invariance under homotopy and stabilization.
• To unify and simplify the computational treatment of persistence invariants for circle-valued maps using linear algebraic structures like linear relations and regular parts of relations.

Proposed method

• The monodromy is defined via linear relations on homology groups of level sets, avoiding the need for infinite cyclic covers or graph representations.
• The algorithm computes the regular part of the composition of linear relations $R(A,B)_{\text{reg}}$, derived from maps between homology groups of adjacent level sets.
• Jordan cells are extracted from the regular part of the monodromy operator $T^{(X,\xi)}(r)$, which is represented as a linear relation $R(\rho)$ associated with a $G_{2m}$-representation.
• The method uses simplicial maps on finite complexes to ensure all angles are weakly regular, enabling algorithmic tractability.
• The construction relies on the decomposition of linear relations into regular and singular parts, with Jordan cells determined by the regular part.
• The algorithm is validated through an explicit example and shown to be shorter than the one in [2], with computational advantages for persistent homology invariants.

Experimental results

Research questions

• RQ1 Can the monodromy of a circle-valued map be proven to be a homotopy invariant without using Novikov homology or infinite cyclic covers?
• RQ2 Is there a more efficient algorithm to compute the Jordan cells of a circle-valued map than the one based on graph representations?
• RQ3 How can Jordan cells be used to compute topological invariants such as Novikov Betti numbers and Alexander polynomials?
• RQ4 What is the relationship between the Jordan cells of a map and the regular part of the associated linear relation?
• RQ5 Can the monodromy be consistently defined and computed for any compact ANR and integral cohomology class via simplicial approximations?

Key findings

• The monodromy operator $T^{(X,\xi)}(r)$ is a homotopy invariant of the pair $(X, \xi)$, proven geometrically without reference to infinite cyclic covers or Novikov homology.
• A new algorithm computes the Jordan cells $\mathcal{J}_r(X;\xi)$ for a simplicial angle-valued map on a finite simplicial complex, with reduced computational complexity compared to prior methods.
• The Jordan cells are equivalent to the regular part of the monodromy relation $R(\rho)_{\text{reg}}$, and their structure is fully captured by the linear relation $R(\rho) = R^\dagger(\beta_m) \cdot R(\alpha_m) \cdots R^\dagger(\beta_1) \cdot R(\alpha_1)$.
• The algorithm computes $R(A,B)_{\text{reg}}$ efficiently, enabling fast calculation of persistence invariants such as barcodes and Jordan cells.
• The method allows for the computation of $L^2$-Betti numbers and Novikov Betti numbers via the Jordan cell decomposition, offering a computational alternative to standard infinite cover constructions.
• The construction is stable under stabilization with acyclic ANRs and respects homotopy equivalences, ensuring robustness in topological data analysis.