Persistence complex

The diagram of the chain complexes and the chain maps for filtration $\{K^{n}\}$ is as follows:

\begin{equation*} \begin{matrix}{} ... & & ... & & ... & & \\ \downarrow ^{\partial } & & \downarrow ^{\partial } & & \downarrow^{\partial } & & \\ C_{n}(K^{1}) & ^{\underrightarrow{~\,i_{\ast }^{1,2}~~}} & C_{n}(K^{2})& ^{\underrightarrow{~\,i_{\ast }^{2,3}~~}} & C_{n}(K^{3}) & ^{\underrightarrow{~\,i_{\ast }^{3,4}~~}} & ... \\ \downarrow ^{\partial } & & \downarrow ^{\partial } & & \downarrow^{\partial } & & \\ C_{n-1}(K^{1}) & ^{\underrightarrow{~\,i_{\ast }^{1,2}~~}} & C_{n-1}(K^{2}) & ^{\underrightarrow{~\,i_{\ast }^{2,3}~~}} & C_{n-1}(K^{3}) & ^{\underrightarrow{~\,i_{\ast }^{3,4}~~}} & ... \\ \downarrow ^{\partial } & & \downarrow ^{\partial } & & \downarrow^{\partial } & & \\ ... & & ... & & ... & & \end{matrix}% \end{equation*}

Observe that all $i_{\ast }^{n,n+1}$ are still inclusions.

Definition. Suppose the homology is over a field $k.$ The chain complex of filtration $\{K^{n}\}$ (related to "persistence complex") is the graded module \begin{equation*} C_{\ast }(\{K^{n}\})=\bigoplus\limits_{n}C_{\ast }(K^{n}) \end{equation*} over the ring of polynomials $k[t]$ with \begin{equation*} t\cdot x=(0,\,i_{\ast }^{1,2}(x_{1}),\,i_{\ast }^{2,3}(x_{1}),...,\,i_{\ast }^{s-1,s}(x_{1})), \end{equation*} where $x=(x_{1},x_{2},...,x_{s})$ and $x_{n}\in C_{\ast }(K^{n}).$

Theorem. Over a field, the homology group of the filtration $\{K^{n}\}$ is isomorphic to the homology group of the chain complex of the filtration, as vector spaces: \begin{equation*} H_{\ast }(\{K^{n}\})\cong H_{\ast }(C_{\ast }(\{K^{n}\})). \end{equation*}

Proof. Should be in [CZ] somewhere.

The computational cost is $O(N^{3})$.