[Paper Review] Discrete Morse Complexes

Manoj K. Chari, Michael Joswig|ArXiv.org|Nov 28, 2000

Topological and Geometric Data Analysis7 references3 citations

TL;DR

This paper introduces the discrete Morse complex, a simplicial complex whose simplices correspond to acyclic matchings in the Hasse diagram of a simplicial complex, thereby formalizing the space of all discrete Morse functions. It establishes connections to combinatorial topology and enumerative combinatorics, proving that the number of perfect Morse matchings on the n-simplex grows at least as fast as a doubly exponential function, with a lower bound derived recursively via cube decomposition.

ABSTRACT

We investigate properties of the set of discrete Morse functions on a simplicial complex as defined by Forman. It is not difficult to see that the pairings of discrete Morse functions of a finite simplicial complex again form a simplicial complex, the discrete Morse complex. It turns out that several known results from combinatorial topology and enumerative combinatorics, which previously seemed to be unrelated, can be re-interpreted in the setting of these discrete Morse complexes.

Motivation & Objective

• To formalize the set of all discrete Morse functions on a simplicial complex as a topological space.
• To study the topological and combinatorial structure of this space, termed the discrete Morse complex.
• To establish connections between discrete Morse theory, graph theory, and enumerative combinatorics.
• To derive tight bounds on the number of perfect Morse matchings in simplicial complexes, especially the n-simplex.
• To explore recursive constructions of acyclic matchings in cube graphs and their implications for Morse function enumeration.

Proposed method

• Represent discrete Morse functions via acyclic matchings in the Hasse diagram of a simplicial complex.
• Define the discrete Morse complex as the abstract simplicial complex formed by all acyclic matchings on the Hasse diagram edges.
• Use graph-theoretic interpretation: each matching edge corresponds to a non-critical pair (face, coface) with decreasing function value.
• Construct perfect acyclic matchings recursively by combining matchings on subcubes of the (n+1)-cube.
• Derive a recursive lower bound r(n+1) = ((n+1)(n-1)/n) * r(n)^2 for the number of perfect Morse matchings on the n-simplex.
• Apply known results on perfect matchings in hypercubes (e.g., Clark, George, and Porter) to compare upper and lower bounds.

Experimental results

Research questions

• RQ1 How can the set of all discrete Morse functions on a simplicial complex be endowed with a natural topological structure?
• RQ2 What is the relationship between discrete Morse complexes and classical combinatorial objects such as rooted forests and perfect matchings?
• RQ3 What are the asymptotic growth rates of the number of perfect Morse matchings in simplicial complexes, particularly the n-simplex?
• RQ4 Can recursive constructions of acyclic matchings in hypercubes yield non-trivial lower bounds on the number of optimal Morse functions?
• RQ5 How do the topological properties of the discrete Morse complex relate to collapsibility and homotopy type of the original complex?

Key findings

• The discrete Morse complex of a simplicial complex Δ is defined as the abstract simplicial complex on the edges of the Hasse diagram, with simplices corresponding to acyclic matchings.
• The pure discrete Morse complex, formed by maximal-dimensional simplices, corresponds to Morse functions with the minimal number of critical faces.
• For the n-simplex, the number of perfect Morse matchings is bounded from below by r(n+1), where r(n+1) = ((n+1)(n-1)/n) * r(n)^2 with r(1)=1, r(2)=2, r(3)=9.
• The lower bound r(n+1) exceeds the product ∏_{k=1}^{n-1} k^{2^{n-k-1}}, indicating a doubly exponential growth rate of approximately (1.289)^{2^n}.
• The recursive construction of matchings using subcubes of the (n+1)-cube yields distinct perfect acyclic matchings, with n+1 choices for the splitting direction, leading to a multiplicative factor in the recurrence.
• The bound r(n+1) is tighter than the upper bound from Clark, George, and Porter’s result on (n+1)-cube perfect matchings, indicating that the recursive construction yields a more efficient lower bound.