Removal and Contraction Operations in $n$D Generalized Maps for Efficient Homology Computation

TL;DR

This paper introduces an efficient algorithm for computing homology generators of nD generalized maps by using removal and contraction operations that preserve homology while simplifying the map.

Contribution

It demonstrates that contraction operations preserve homology under certain conditions and develops a method to simplify nD generalized maps for faster homology computation.

Findings

Contraction operations preserve homology under specific conditions.

The proposed algorithm significantly reduces the number of cells while maintaining homology.

The method improves efficiency in computing homology generators for nD generalized maps.

Abstract

In this paper, we show that contraction operations preserve the homology of $n$D generalized maps, under some conditions. Removal and contraction operations are used to propose an efficient algorithm that compute homology generators of $n$D generalized maps. Its principle consists in simplifying a generalized map as much as possible by using removal and contraction operations. We obtain a generalized map having the same homology than the initial one, while the number of cells decreased significantly. Keywords: $n$D Generalized Maps; Cellular Homology; Homology Generators; Contraction and Removal Operations.