Computing Homological Information Based on Directed Graphs within Discrete Objects

Abstract

n-dimensional discrete objects can be interpreted as cubical complexes which are suitable for the study of their homology groups in order to understand the original discrete object. The classic approach consists in computing the Normal Smith Form of some matrices associated to the cubical complex. Further approaches deal mainly with a pre-processing of the matrices in order to reduce their size. In this paper we propose a new approach, initially based on discrete Morse theory, which computes some homological information (Betti numbers and representative cycles) without calculating the Normal Smith Form. It works on any dimension, and it can also be applied to any kind of regular cell complex.