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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.^[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.

Herbert Edelsbrunner develops algorithms for computational geometry and computational topology. His research account describes a progression from data structures and geometry to topological methods, guided by applications.^[ref] He coauthored Computational Topology: An Introduction by Herbert Edelsbrunner and John L. Harer with John L. Harer.^[ref]

A central computational challenge is to replace a continuous shape by finite combinatorial data while preserving the invariant being studied. For example, a triangulated surface can be described by vertices, edges and faces, allowing homology to be computed from boundary relations. The representation supplies an algorithmic route to a geometric question; the quality of the input model remains part of interpreting the answer.

John L. Harer works at the intersection of topology, geometry and computational data analysis. His Duke research description emphasizes geometric, combinatorial and computational methods for analyzing shape and data.^[ref] With Herbert Edelsbrunner, he coauthored Computational Topology: An Introduction by Herbert Edelsbrunner and John L. Harer, connecting topological invariants to computational methods.^[ref]

An elementary illustration is a triangular outline: it has one connected component and one unfilled loop. Filling its interior keeps the connected component but removes the hole. Homology distinguishes these cases, showing why a description of shape can capture something different from coordinates or pairwise distances. The example illustrates the subject rather than attributing a particular theorem or discovery to the book's authors.