[Paper Review] Graph Representations and Topology of Real and Angle Valued Maps

Dan Burghelea, Stefan Haller|arXiv (Cornell University)|Feb 6, 2012

Topological and Geometric Data Analysis9 references3 citations

TL;DR

This paper introduces graph representations and topological invariants—bar codes and Jordan cells—for real and angle-valued tame maps, proving the homotopy invariance of the sum of closed and open bar codes in adjacent dimensions and of the monodromy (Jordan cells). It reinterprets Morse–Novikov theory via critical values and persistence invariants, offering a broader framework than classical gradient-based approaches.

ABSTRACT

In this paper we review the definition of the invariants "bar codes" and "Jordan cells" of real and angle valued tame maps as proposed in Burghelea and Dey and Carlsson et al and prove the homotopy invariance of the sum # B^c_r +#B^o_{r-1}$ and of the Jordan cells. Here B^c_r resp. B^o_r denote the sets of closed resp. open bar codes in dimension r and # denotes cardinality. In addition we provide calculation of some familiar topological invariants in terms of bar codes and Jordan cells. The presentation provides a different perspective on Morse-Novikov theory based on critical values, bar codes and Jordan cells rather than on critical points instantons and closed trajectories of a gradient of a real or angle valued map.

Motivation & Objective

• To establish a new topological framework for real and angle-valued maps using bar codes and Jordan cells as invariants.
• To prove that the sum $\sharp\mathcal{B}^{c}_{r} + \sharp\mathcal{B}^{o}_{r-1}$ and the set of Jordan cells are homotopy invariants of the pair $(X, \xi_f)$, where $\xi_f \in H^1(X; \mathbb{Z})$.
• To provide a reformulation of Morse–Novikov theory based on critical values, bar codes, and monodromy instead of critical points and gradient trajectories.
• To extend persistence-theoretic invariants to continuous maps with compact ANR fibers, generalizing results beyond tame maps.
• To offer a computational method for bar codes and Jordan cells using elementary transformations on graph representations.

Proposed method

• Define bar codes as intervals $[a,b]$, $(a,b)$, or mixed $[a,b)$, $(a,b]$ in the real line, representing persistence of homology classes in fibers of a tame map.
• Define Jordan cells as indecomposable blocks of linear automorphisms on finite-dimensional vector spaces over a field $\kappa$, capturing monodromy in angle-valued maps.
• Use graph representations to encode the structure of level sets and transition maps between them, enabling computation of invariants via elementary transformations.
• Apply the canonical long exact sequence associated with a tame map to relate homology of sublevel sets to bar codes and Jordan cells.
• Utilize elementary transformations $T_i(j)$ to simplify representations and eliminate bar codes step-by-step, tracking invariants through the process.
• Prove homotopy invariance by showing that the number $N_r(f) = \sharp\mathcal{B}^{c}_{r}(f) + \sharp\mathcal{B}^{o}_{r-1}(f)$ and the monodromy $(V_r(f), T_r(f))$ remain unchanged under homotopy equivalences preserving the cohomology class $\xi_f$.

Experimental results

Research questions

• RQ1 How can bar codes and Jordan cells be defined as invariants for real and angle-valued tame maps?
• RQ2 What is the homotopy invariance status of the sum $\sharp\mathcal{B}^{c}_{r} + \sharp\mathcal{B}^{o}_{r-1}$ and the set of Jordan cells?
• RQ3 Can Morse–Novikov theory be reinterpreted using critical values and persistence invariants rather than critical points and gradient trajectories?
• RQ4 How do bar codes and Jordan cells relate to classical topological invariants like Betti numbers and homology groups?
• RQ5 Can the framework be extended to continuous maps with compact ANR fibers, not just tame maps?

Key findings

• The sum $\sharp\mathcal{B}^{c}_{r}(f) + \sharp\mathcal{B}^{o}_{r-1}(f)$ is a homotopy invariant of the pair $(X, \xi_f)$, where $\xi_f \in H^1(X; \mathbb{Z})$ is the cohomology class of the map $f$.
• The set of Jordan cells $\mathcal{J}_r(f)$ is also a homotopy invariant of the pair $(X, \xi_f)$, with the monodromy $ (V_r(f), T_r(f)) $ preserved under homotopy equivalence.
• For the example in Section 7, the bar codes include $[2,3]$, $(4,5)$, and $(\theta_6, \theta_1 + 2\pi]$, while the only Jordan cell is $(2,2)$ in dimension 1.
• The monodromy in dimension 1 is identified as the regular part of the linear relation defined by the maps $\omega_1$ and $\omega_2$, yielding the Jordan cell $(2,2)$.
• Elementary transformations $T_i(j)$ allow the decomposition of a graph representation into irreducible components, enabling the systematic elimination of bar codes and extraction of invariants.
• In dimension 0, the only invariant is the Jordan cell $\rho^{II}(1;1)$, reflecting the connectedness of all fibers in that dimension.