A combinatorial method to compute explicit homology cycles using Discrete Morse Theory

In this paper we shall describe a combinatorial method related to Discrete Morse Theory, which allows us to calculate explicit homology cycles in polyhedral complexes. These cycles form a basis, in the case when the critical cells are in an isolated dimension. We illustrate the use of this technique by several examples from combinatorial topology, including the complexes of multihomomorphisms between complete graphs. Our method is optimal from the computational complexity point of view, requiring execution time which is linear in the number of d -cells.