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Stratified Morse Theory for Cell Complexes

Vidit Nanda, Francesca Tombari

TL;DR

The paper develops a stratified version of discrete Morse theory for finite regular CW complexes by introducing halos and shadows to handle the closure issue and by enforcing stratum-wise Forman conditions. It proves two Morse-type lemmas: (A) sublevel sets do not change across regular intervals when no $s$-critical values are present, via filtration-preserving collapses, and (B) across an $s$-critical value, the local change is captured by a shadow-to-closure attachment ${\mathbf{sh}}(\sigma;f) \hookrightarrow {\mathrm{cl}}(\sigma)$, with $s$-critical cells distinguished from ordinary critical ones. To obtain a simplicial analogue of the tangential-normal data, the authors lift the construction to the barycentric subdivision using the upper envelope ${\mathbf U}f$, obtaining a horizontal-vertical join decomposition $H \star V$ of the local Morse data, where $H$ lies in lower strata and $V$ in higher strata. The work connects to filtered discrete Morse theory, stratified multivector fields, and Conley indices, while offering a concrete, combinatorial framework with explicit halo-collapsibility that avoids manifold regularity assumptions and conical-neighborhood requirements.

Abstract

We develop a version of discrete Morse theory for finite regular CW complexes equipped with an auxiliary stratification. The key construction is the halo of a cell, which contains all those faces in the boundary that enter closed sublevelsets precisely when the threshold reaches that cell's value. The complement of this halo in the boundary, called the shadow, is always a subcomplex. A stratified discrete Morse function requires Forman's conditions on each stratum together with the requirement that closures of paired cells admit filtered collapses onto their shadows. We establish fundamental Morse lemmas: filtered collapses across regular intervals, and controlled attachments at critical values. For functions satisfying only the stratum-wise Forman condition, we construct an upper envelope on the barycentric subdivision whose local Morse data decomposes into horizontal and vertical components. This yields a simplicial analogue of the standard tangential-normal splitting of local Morse data in the sense of Goresky and MacPherson.

Stratified Morse Theory for Cell Complexes

TL;DR

The paper develops a stratified version of discrete Morse theory for finite regular CW complexes by introducing halos and shadows to handle the closure issue and by enforcing stratum-wise Forman conditions. It proves two Morse-type lemmas: (A) sublevel sets do not change across regular intervals when no -critical values are present, via filtration-preserving collapses, and (B) across an -critical value, the local change is captured by a shadow-to-closure attachment , with -critical cells distinguished from ordinary critical ones. To obtain a simplicial analogue of the tangential-normal data, the authors lift the construction to the barycentric subdivision using the upper envelope , obtaining a horizontal-vertical join decomposition of the local Morse data, where lies in lower strata and in higher strata. The work connects to filtered discrete Morse theory, stratified multivector fields, and Conley indices, while offering a concrete, combinatorial framework with explicit halo-collapsibility that avoids manifold regularity assumptions and conical-neighborhood requirements.

Abstract

We develop a version of discrete Morse theory for finite regular CW complexes equipped with an auxiliary stratification. The key construction is the halo of a cell, which contains all those faces in the boundary that enter closed sublevelsets precisely when the threshold reaches that cell's value. The complement of this halo in the boundary, called the shadow, is always a subcomplex. A stratified discrete Morse function requires Forman's conditions on each stratum together with the requirement that closures of paired cells admit filtered collapses onto their shadows. We establish fundamental Morse lemmas: filtered collapses across regular intervals, and controlled attachments at critical values. For functions satisfying only the stratum-wise Forman condition, we construct an upper envelope on the barycentric subdivision whose local Morse data decomposes into horizontal and vertical components. This yields a simplicial analogue of the standard tangential-normal splitting of local Morse data in the sense of Goresky and MacPherson.
Paper Structure (8 sections, 16 theorems, 38 equations, 2 figures)

This paper contains 8 sections, 16 theorems, 38 equations, 2 figures.

Table of Contents

  1. Stratified Cell Complexes
  2. Stratifications
  3. Filtered Collapses
  4. Halos and Shadows
  5. Stratified Discrete Morse Functions
  6. Local Morse Data
  7. Local Morse Data via the Upper Envelope
  8. Multivector Fields and the Conley Index

Key Result

Theorem 1

Let $[c, d] \subset {\mathbb R}$ be an interval whose preimage $f^{-1}\left([c,d]\right)$ is $\left\{{\sigma}\right\}$. If $\sigma$ is not s-critical, then ${\mathrm{\bf cl}}(f_{\leq d})$ admits a filtration-preserving collapse onto ${\mathrm{\bf cl}}(f_{\leq c})$.

Figures (2)

  • Figure 1: A piece of a cell complex containing one vertex and four edges, with $f$-values indicated on the cells. Note that the closure of $f_{\leq 1}$ already contains the central vertex, even though that vertex has $f$-value $3$.
  • Figure 2: A subdivided version of the Example from Figure \ref{['fig:closure']}, this time with ${\mathbf{U}{f}}$-values on simplices; here the central vertex does contribute to the change in topology because its lower link consists of two vertices. These lie at the barycenters of the original edges, and hence have ${\mathbf{U}{f}}$-values $1$ and $2$.

Theorems & Definitions (40)

  • Theorem : A
  • Theorem : B
  • Theorem : C
  • Remark 1.1
  • Proposition 1.2
  • proof
  • Proposition 1.3
  • proof
  • Remark 1.4
  • Remark 1.5
  • ...and 30 more