We summarize the work so far and relate it to previous results. Our input is a filtered complex and we wish to find its th homology . In each dimension the homology of complex becomes a vector space over a field, described fully by its rank . (Over a field , is a -module which is a vector space.)

We need to choose compatible bases across the filtration (compatible bases for and ) in order to compute persistent homology for the entire filtration. Hence, we form the persistence module corresponding to , which is a direct sum of these vector spaces (). By the structure theorem, a basis exists for this module that provides compatible bases for all the vector spaces.

Specifically, each -interval describes a basis element for the homology vector spaces starting at time until time . This element is a -cycle that is completed at time , forming a new homology class. It also remains non-bounding until time , at which time it joins the boundary group .

A natural question is to ask when is a basis element for the persistent groups . Recall the equation Since for all , hence for . The three inequalities define a triangular region in the index-persistence plane, as shown in Figure below.

The triangular region gives us the values for which the -cycle is a basis element for . This is known as the -triangle Lemma:

Let be the set of triangles defined by -intervals for the -dimensional persistence module. The rank of is the number of triangles in containing the point .

Hence, computing persistent homology over a field is equivalent to finding the corresponding set of -intervals.

Source: “Computing Persistent Homology” by Zomorodian and Carlsson

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