Homology

Computing simplicial homology based on efficient Smith normal form algorithms Geometric properties of topological spaces are conveniently expressed by algebraic invariants of the space. This paper focuses on methods for the computer calculation of the homology of finite simplicial complexes and its applications. The calculation of homology with integer coefficients of a simplicial complex reduces to the calculation of the Smith Normal Form of the boundary matrices which, in general, are sparse. First, the authors provide a review of several algorithms for the calculation of the Smith Normal Form of sparse matrices and compare their running times for actual boundary matrices, then they describe alternative approaches to the calculation of simplicial homology. In the last section they present motivating examples and actual experiments with the GAP package (implemented by the authors). There is an example with calculations of Lie algebra homology.