[Paper Review] Combinatorial realization of the Thom-Smale complex via discrete Morse theory

Étienne Gallais|ArXiv.org|Mar 18, 2008

Topological and Geometric Data Analysis11 references3 citations

TL;DR

This paper establishes a combinatorial realization of the Thom-Smale complex for smooth manifolds using Forman's discrete Morse theory, proving that any smooth Morse function's Thom-Smale complex can be realized via a triangulation and a discrete Morse function. The key contribution is a constructive proof that any Euler structure on a smooth oriented closed 3-manifold admits a complete matching realization on the Hasse diagram of a triangulation, linking discrete Morse theory to spin^c structures and Heegaard-Floer homology.

ABSTRACT

In the case of smooth manifolds, we use Forman's discrete Morse theory to realize combinatorially any Thom-Smale complex coming from a smooth Morse function by a couple triangulation-discrete Morse function. As an application, we prove that any Euler structure on a smooth oriented closed 3-manifold has a particular realization by a complete matching on the Hasse diagram of a triangulation of the manifold.

Motivation & Objective

• To establish a combinatorial realization of the Thom-Smale complex arising from a smooth Morse function using discrete Morse theory.
• To provide two direct proofs of the fundamental identity ∂^V ∘ ∂^V = 0 in the combinatorial Thom-Smale complex, one geometric and one algebraic.
• To demonstrate that the combinatorial Thom-Smale complex is homotopy equivalent to the simplicial chain complex.
• To show that any Euler structure on a smooth oriented closed 3-manifold can be realized via a complete matching on the Hasse diagram of a triangulation.
• To connect discrete Morse theory to spin^c structures and Heegaard-Floer homology by realizing both the pointed Heegaard splitting and spin^c structures combinatorially.

Proposed method

• Use Forman’s discrete Morse theory on finite simplicial complexes, modeling Morse functions via combinatorial vector fields (Morse matchings) on the Hasse diagram.
• Define the combinatorial Thom-Smale complex (C^V_*, ∂^V) using critical cells and V-paths, with coefficients determined by incidence numbers and path multiplicities in {±1}.
• Prove ∂^V ∘ ∂^V = 0 via a geometric argument based on path cancellation and an algebraic argument relying on chain homotopy equivalence to the simplicial chain complex.
• Construct a C^1-triangulation of a smooth manifold and a discrete Morse vector field such that the resulting combinatorial Thom-Smale complex matches the smooth Thom-Smale complex.
• Use the correspondence between Euler structures and spin^c structures on 3-manifolds to realize any Euler structure via a complete matching on the Hasse diagram of a triangulation.
• Leverage results from Heegaard-Floer homology and Turaev’s correspondence between spin^c structures and Euler structures to embed the combinatorial realization into the broader topological framework.

Experimental results

Research questions

• RQ1 Can every smooth Thom-Smale complex be combinatorially realized as a discrete Morse complex via a triangulation and a discrete Morse function?
• RQ2 What is the geometric and algebraic justification for the identity ∂^V ∘ ∂^V = 0 in the combinatorial Thom-Smale complex?
• RQ3 Is there a constructive method to realize any Euler structure on a smooth oriented closed 3-manifold through a complete matching on the Hasse diagram of a triangulation?
• RQ4 How does the combinatorial realization of the Thom-Smale complex relate to spin^c structures and Heegaard-Floer homology?
• RQ5 Can the gradient vector field of a Morse function be homotoped to a vector field induced by a discrete Morse matching on a triangulation?

Key findings

• The paper provides two proofs of ∂^V ∘ ∂^V = 0: one geometric, based on cancellation of non-stationary closed V-paths, and one algebraic, showing that the combinatorial Thom-Smale complex is chain homotopy equivalent to the simplicial chain complex.
• Any smooth Morse function on a manifold induces a combinatorial Thom-Smale complex that is realizable via a triangulation and a discrete Morse function, establishing a full combinatorial realization of the smooth complex.
• For any smooth oriented closed 3-manifold, every Euler structure arises from a complete matching on the Hasse diagram of some triangulation, proving a combinatorial realization of Euler structures.
• The existence of such triangulations with complete matchings is guaranteed, and the realization is compatible with the correspondence between spin^c structures and Euler structures.
• The construction ensures that the vector field associated to the matching is homologous to the Morse-Smale gradient vector field, validating the topological fidelity of the combinatorial model.
• The results support the combinatorial foundation of Heegaard-Floer homology by showing that both the pointed Heegaard splitting and spin^c structures can be realized combinatorially via discrete Morse data.