Efficient algorithmic computation of homology groups and the fundamental
group of a subset of Euclidean space is among the basic problems
in present-day Applied Topology. Particular interest is in algorithms
taking finite CW complexes on input. At the end of 20th century
Robin Forman proposed a version of the classical
Morse theory for a CW complex embedded with a combinatorial counterpart of
a vector field. In the talk we will demonstrate how this theory may be
fruitfully used in the construction of homology algorithms.