Homology of Finite Topological Spaces

TL;DR

The paper introduces a new polynomial-time algorithm for computing the homology groups of finite T0-spaces using irreducible cycles and the orb complex, simplifying calculations in algebraic topology.

Contribution

It presents a novel method leveraging irreducible cycles and the orb complex to efficiently compute homology groups of finite T0-spaces, improving computational approaches.

Findings

Algorithm computes homology groups in polynomial time.

Orb complex provides a smaller, effective tool for homology computation.

Kernel analysis of the homology map enables efficient calculations.

Abstract

A new method is given for computing generators of the homology groups with integer coefficients for any finite $T_0$-space. An important role in this method is played by irreducible cycles which are defined here and give rise to continuous injective maps between spaces called immersions. The so-called orb complex which is much smaller than the order complex induces a surjective map of its homology to the simplicial homology of the space. An analysis of the kernel of this map allows to define an effective algorithm for computing the homology groups, whose time complexity is polynomial in the size of the space.