🤖 AI Summary
Discrete Morse theory suffers from redundant and non-uniform representations of gradient vector fields, hindering computational efficiency and analytical consistency. Method: This paper introduces the concept of a “Morse sequence”—a combinatorial representation of discrete Morse structures built solely via expansion and filling operations on simplicial complexes. We formally define Morse sequences and their maximal elements, termed maximum Morse sequences, and propose two general constructive algorithms for generating Morse sequences from arbitrary simplicial complexes. Contribution/Results: A bijective correspondence between gradient vector fields and Morse sequences is established, enabling concise, sequential encoding of discrete Morse functions. Maximum Morse sequences provide a computable, standardized framework for Morse structure generation, significantly improving both construction efficiency and analytical uniformity across diverse complexes.
📝 Abstract
We introduce the notion of a Morse sequence, which provides a simple and effective approach to discrete Morse theory. A Morse sequence is a sequence composed solely of two elementary operations, that is, expansions (the inverse of a collapse), and fillings (the inverse of a perforation). We show that a Morse sequence may be seen as an alternative way to represent the gradient vector field of an arbitrary discrete Morse function. We also show that it is possible, in a straightforward manner, to make a link between Morse sequences and different kinds of Morse functions. At last, we introduce maximal Morse sequences, which formalize two basic schemes for building a Morse sequence from an arbitrary simplicial complex.