Summary of Persistent Homology

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

We need to choose compatible bases across the filtration (compatible bases for $H_k^i$ and $H_k^{i+p}$) in order to compute persistent homology for the entire filtration. Hence, we form the persistence module $\mathscr{M}$ corresponding to $K$, which is a direct sum of these vector spaces ($\alpha(\mathscr{M})=\bigoplus M^i$). By the structure theorem, a basis exists for this module that provides compatible bases for all the vector spaces.

Specifically, each $\mathcal{P}$-interval $(i,j)$ describes a basis element for the homology vector spaces starting at time $i$ until time $j-1$. This element is a $k$-cycle $e$ that is completed at time $i$, forming a new homology class. It also remains non-bounding until time $j$, at which time it joins the boundary group $B_k^j$.

A natural question is to ask when $e+B_k^l$ is a basis element for the persistent groups $H_k^{l,p}$. Recall the equation $\displaystyle H_k^{i,p}=Z_k^i/(B_k^{i+p}\cap Z_k^i).$ Since $e\notin B_k^l$ for all $l, hence $e\notin B_k^{l+p}$ for $l+p. The three inequalities $\displaystyle l+p define a triangular region in the index-persistence plane, as shown in Figure below.

The triangular region gives us the values for which the $k$-cycle $e$ is a basis element for $H_k^{l,p}$. This is known as the $k$-triangle Lemma:

Let $\mathcal{T}$ be the set of triangles defined by $\mathcal{P}$-intervals for the $k$-dimensional persistence module. The rank $\beta_k^{l,p}$ of $H_k^{l,p}$ is the number of triangles in $\mathcal{T}$ containing the point $(l,p)$.

Hence, computing persistent homology over a field is equivalent to finding the corresponding set of $\mathcal{P}$-intervals.

Source: “Computing Persistent Homology” by Zomorodian and Carlsson