Morse shellings out of discrete Morse functions

TL;DR

This paper establishes a correspondence between discrete Morse functions and Morse shellings on barycentric subdivisions of simplicial complexes, linking topological handle decompositions with combinatorial structures.

Contribution

It proves that discrete Morse functions induce Morse shellings on barycentric subdivisions, preserving critical face indices and extending classical shellings.

Findings

Discrete Morse functions induce Morse shellings on second barycentric subdivisions.

Critical tiles in Morse shellings correspond to critical faces of Morse functions.

The correspondence holds for smooth Morse functions on manifolds after subdivisions.

Abstract

From the topological viewpoint, Morse shellings of finite simplicial complexes are {\it pinched} handle decompositions and extend the classical shellings. We prove that every discrete Morse function on a finite simplicial complex induces Morse shellings on its second barycentric subdivision whose critical tiles-or pinched handles-are in oneto-one correspondence with the critical faces of the function, preserving the index. The same holds true, given any smooth Morse function on a closed manifold, for any piecewise-linear triangulation on it after sufficiently many barycentric subdivisions.