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Computing persistent homology. (English) Zbl 1069.55003
Summary: We show that the persistent homology of a filtered d-dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enables us to derive a natural algorithm for computing persistent homology of spaces in arbitrary dimension over any field. This result generalizes and extends the previously known algorithm that was restricted to subcomplexes of \(\mathbb S^3\) and \(\mathbb Z_2\) coefficients. Finally, our study implies the lack of a simple classification over non-fields. Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary principal ideal domain in any dimension.

MSC:
55N35 Other homology theories in algebraic topology
57Q99 PL-topology
55-04 Software, source code, etc. for problems pertaining to algebraic topology
55U10 Simplicial sets and complexes in algebraic topology
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
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