= Computational topology
<Computational topology> develops <algorithms> for topological questions about finite representations of spaces, such as <simplicial complexes>. It connects discrete <data structures> to invariants that ignore irrelevant geometric distortion. <connected components> and <homology groups> are examples of computable structural information.<Computational Topology: An Introduction by Herbert Edelsbrunner and John L. Harer>[publisher description, topology through algorithms]{ref}
For a small <simplicial homology> calculation over a <field>, take three vertices joined by three edges in a cycle, without a filled face. The matrix of the first <boundary operator> has <matrix rank> two, so this <linear map> has a <kernel of a linear map> of <dimension> one; with no two-dimensional face, that cycle is not a boundary. Thus the first <Betti number> is one. Adding the filled triangular face makes the same cycle a boundary and reduces the first <Betti number> to zero. The vertex coordinates need not change: the encoded incidence structure determines whether the hole is present.
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