Homology of Finite Topological Spaces

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.