The discrete flow category: structure and computation

TL;DR

The paper develops a discrete flow category Flo_Σ[X] from discrete Morse functions on regular CW complexes and proves that its classifying space recovers the homotopy type of the underlying complex. It establishes that, for discrete Morse functions on simplicial complexes, the Hom posets are CW posets, enabling a CW realization of Hom sets and simplifying topological interpretations. An algorithm (with Python implementation) computes Hom posets efficiently, and the paper proves a key spectral-sequence result: the spectral sequence of the double nerve collapses on page 2, facilitating homology computations from discrete Morse data. The work connects discrete Morse theory with simplicial-set techniques, introduces generalized collapses and unique factorization categories, and provides concrete examples to illustrate the methods on standard spaces like S^2, D^3, and T^2, with implications for topological data analysis and combinatorial topology.

Abstract

In this article, we use concepts and methods from the theory of simplicial sets to study discrete Morse theory. We focus on the discrete flow category introduced by Vidit Nanda, and investigate its properties in the case where it is defined from a discrete Morse function on a regular CW complex. We design an algorithm to efficiently compute the Hom posets of the discrete flow category in this case. Furthermore, we show that in the special case where the discrete Morse function is defined on a simplicial complex, then each Hom poset has the structure of a face poset of a regular CW complex. Finally, we prove that the spectral sequence associated to the double nerve of the discrete flow category collapses on page 2.

Figures (21)

• Figure 1: The barycentric subdivision of the simplicial complex $\Delta^2$.
• Figure 2: An example of a discrete Morse function. The critical values are underlined and colored red.
• Figure 3: The gradient vector field for the discrete Morse function in \ref{['fig:dmf_example']}.
• Figure 4: An elementary collapse.
• Figure 5: The simplicial set $\mathop{\mathrm{diag}}\nolimits(\mathcal{N} \circ \mathscr{S})$. Note that the top and bottom edges are degenerate, as $s_0 \mathop{\mathrm{id}}\nolimits_a = (s_0, s_0) a$, and similar for $s_0 \mathop{\mathrm{id}}\nolimits_b$. The left and right edges are, however, not degenerate as 1--simplices in $\mathop{\mathrm{diag}}\nolimits(\mathcal{N} \circ \mathscr{S})$.

Theorems & Definitions (93)

• Theorem A
• Theorem B
• Theorem C
• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Corollary 2.4
• Definition 3.1
• Definition 3.2
• Definition 3.3