Cech Type Approach to Computing Homology of Maps

A new approach to algorithmic computation of the homology of spaces and maps is presented. The key point of the approach is a change in the representation of sets. The proposed representation is based on a combinatorial variant of the Čech homology and the Nerve Theorem. In many situations this change of the representation of the input may help in bypassing the problems with the complexity of the standard homology algorithms by reducing the size of nedcessary input. We show that the approach is particularly advantegous in the case of homology map algorithms.