[Paper Review] Polyhedral Representation of Discrete Morse Functions on Regular CW Complexes and Posets

Ethan D. Bloch|arXiv (Cornell University)|Aug 22, 2010

Topological and Geometric Data Analysis6 references3 citations

TL;DR

This paper establishes a direct correspondence between discrete Morse functions on finite regular CW complexes and polyhedral Morse functions on embedded barycentric subdivisions. It proves that every discrete-critical cell (in Forman's sense) corresponds exactly to a polyhedral-critical vertex (in Banchoff's sense) under an appropriate embedding in Euclidean space, preserving criticality via projection onto a generic line.

ABSTRACT

It is proved that every discrete Morse function in the sense of Forman on a finite regular CW complex can be represented by a polyhedral Morse function in the sense of Banchoff on an appropriate embedding in Euclidean space of the barycentric subdivision of the CW complex; such a representation preserves critical points. The proof is stated in terms of discrete Morse functions on a class of posets that is slightly broader than the class of face posets of finite regular CW complexes.

Motivation & Objective

• To bridge the conceptual gap between Forman’s discrete Morse theory and Banchoff’s polyhedral Morse theory on finite regular CW complexes.
• To resolve the lack of a rigorous connection between the two widely cited but seemingly disparate Morse-theoretic frameworks.
• To show that critical cells in Forman’s discrete setting can be geometrically realized as critical vertices in Banchoff’s polyhedral setting.
• To establish that the barycentric subdivision of a regular CW complex provides a canonical simplicial structure where cell criticality is preserved under projection.
• To demonstrate that for any discrete Morse function, there exists a sufficiently high-dimensional Euclidean embedding such that criticality is preserved under generic linear projection.

Proposed method

• Use the barycentric subdivision of a finite regular CW complex to convert cells into vertices of a simplicial complex.
• Embed the resulting simplicial complex into Euclidean space using a map that preserves the discrete Morse function values as coordinates along the x-axis.
• Apply Banchoff’s vertex index formula based on the sign of the sum of (-1)^dim(s) over simplices containing a vertex where the projection is maximal.
• Leverage poset-theoretic properties of the face poset of the CW complex to relate the discrete Morse function’s criticality condition to the polyhedral index.
• Use the fact that for a 2-wide poset, the discrete Morse condition implies that either a unique predecessor or a unique successor has value ≥ or ≤ the cell’s value, respectively.
• Prove that the polyhedral index is non-zero (i.e., vertex is polyhedral-critical) if and only if the corresponding cell is discrete-critical under the function.

Experimental results

Research questions

• RQ1 Can Forman’s discrete Morse functions on finite regular CW complexes be geometrically realized via Banchoff’s polyhedral Morse theory?
• RQ2 Is there a canonical embedding of the barycentric subdivision of a regular CW complex into Euclidean space such that critical cells correspond to critical vertices under linear projection?
• RQ3 Does the criticality of a cell in Forman’s sense correspond exactly to the non-vanishing of Banchoff’s index at the corresponding barycenter?
• RQ4 Can the discrete Morse function’s criticality condition be characterized purely in terms of the poset structure and the projection-induced ordering on the barycentric subdivision?
• RQ5 What is the precise relationship between the poset-theoretic properties of the face poset and the topological invariants (indices) in Banchoff’s framework?

Key findings

• Every discrete Morse function on a finite regular CW complex admits a polyhedral realization via an embedding of its barycentric subdivision into Euclidean space.
• For any sufficiently large dimension m, there exists an embedding of the barycentric subdivision into R^m such that the critical cells of the discrete Morse function correspond exactly to the vertices with non-zero Banchoff index.
• The polyhedral index a(φ(b), ξ) is non-zero if and only if the corresponding cell b is discrete-critical, and in that case, |a(φ(b), ξ)| = 1, with sign determined by the rank of the cell.
• When a cell is discrete-ordinary, the polyhedral index is zero, corresponding to the existence of a unique predecessor or successor with value ≥ or ≤ the cell’s value.
• The construction relies on the poset structure of the face poset and the 2-wide property to ensure that the index computation depends only on the local behavior of the discrete Morse function.
• The result holds for any unit vector ξ in S^{m-1} after an appropriate rotation and translation of the embedding, ensuring genericity of the projection.