Stratified Morse Theory for Cell Complexes

TL;DR

This paper extends discrete Morse theory to stratified cell complexes, introducing new concepts like halos and shadows to analyze the topology of stratified spaces with Morse functions.

Contribution

It develops a stratified version of discrete Morse theory, including new constructions and lemmas, for finite regular CW complexes with stratifications.

Findings

Established fundamental Morse lemmas for stratified complexes.

Constructed an upper envelope on barycentric subdivisions for Morse data.

Decomposed local Morse data into horizontal and vertical components.

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.