On homology of finite topological spaces

TL;DR

This paper introduces a new spectral sequence-based method for computing homology groups of finite topological spaces and posets, simplifying calculations and enabling various applications in topology and combinatorics.

Contribution

It presents a novel, efficient spectral sequence approach for homology computation of finite spaces, extending classical theorems and providing new formulas.

Findings

Simplified homology computation with fewer calculations

Generalization of Hurewicz theorem for CW-complexes

New formulas for M"obius function of posets

Abstract

We develop a new method to compute the homology groups of finite topological spaces (or equivalently of finite partially ordered sets) by means of spectral sequences giving a complete and simple description of the corresponding differentials. Our method proves to be powerful and involves far fewer computations than the standard one. We derive many applications of our technique which include a generalization of Hurewicz theorem for regular CW-complexes, results in homological Morse theory and formulas to compute the M\"obius function of posets.