  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.
