Homology computation by reduction of chain complexes

Computation of cubical homology, cohomology, and (co)homological operations via chain contraction

“…These reductions correspond e.g. to collapsing external faces or to joining adjacent cells, which were a basis for an algorithm for homology computation by reduction of chain complexes introduced in [33]. Its generalization was implemented in [9] (see also [34]) and is used in the software package available at the project's website [47].…”

“…Moreover, if the coefficient ring is a field then SNF can be computed in cubic time using the simple method based on Gaussian elimination (see e.g. [33]). …”

Coreduction Homology Algorithm

“…As in [13] we say that a pair (a, b) of elements of S is a reduction pair if κ(b, a) is invertible in R. A reduction pair (a, b) is said to be an elementary reduction pair if cbd S a = {b}. In this case we will also say that a is a free face in S. Similarly, we define an elementary coreduction pair as a reduction pair (a, b) such that bd S b = {a} and in this case we call b a free coface in S. Proof: First assume that (a, b) is an elementary reduction pair.…”

“…The methods of chain complex reduction, originally proposed in [13] and then developed in [15,12,18] constitute such an approach. They consist in iterating the process of replacing the chain complex, or even better some combinatorial representation of the topological space, by a smaller one with the same homology and computing the homology only when no more reductions are possible.…”

“…The algorithm presented in [13] is a modification of Smith diagonalization for sparse matrices. The modification consists in removing columns and rows of generators with trivial homology in all relevant matrices of boundary maps as soon as the triviality is verified.…”