A Divide-And-Conquer Method for computing the Betti numbers of Finite Topological Spaces

TL;DR

This paper introduces a divide-and-conquer algorithm leveraging the Mayer-Vietoris sequence to efficiently compute Betti numbers of finite T0-spaces, with analysis of its parallelization potential.

Contribution

It presents a novel divide-and-conquer approach for Betti number computation in finite T0-spaces using Mayer-Vietoris, including parallelization analysis.

Findings

Algorithm effectively computes Betti numbers for finite T0-spaces.

Parallelization potential reduces computational costs.

Method outperforms traditional approaches in specific cases.

Abstract

A divide-and-conquer algorithm for computing the Betti numbers of finite $T_0$-spaces is presented. It extensively uses the Mayer-Vietoris sequence for open coverings. In the end, the computational costs for a parallelisation of this method are given.