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Homology groups of filtrations

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We provide formal definitions now. All cell complexes are finite.

Suppose we have a one-parameter filtration: \begin{equation*} K^{1}\hookrightarrow K^{2}\hookrightarrow K^{3}\hookrightarrow \ldots \ \hookrightarrow K^{s}. \end{equation*} Here $K^{1},K^{2},\ldots ,K^{s}$ are cell complexes and the arrows represent the inclusions $i^{n,n+1}:K^{n}\hookrightarrow K^{n+1}$ and so do $i^{nm}:K^{n}\hookrightarrow K^{m},n\leq m$.

We will denote the filtration by $$\{K^{n},i^{nm}:n,m=1,2,...,s,n\leq m\},$$ or simply $\{K^{n}\}.$

Next, homology generates a "direct system" of groups and homomorphisms: \begin{equation*} H_{\ast }(K^{1})\rightarrow H_{\ast }(K^{2})\rightarrow \ldots \ \rightarrow H_{\ast }(K^{s})\longrightarrow 0. \end{equation*} We denote this direct system by $$\{H_{\ast }(K^{n}),i_{\ast}^{nm}:n,m=1,2,...,s,n\leq m\},$$ or simply $\{H_{\ast }(K^{n})\}.$ The zero is added in the end for convenience.

Our goal is to define a single structure that captures all homology classes in the whole filtration without double counting. The rationale is that if $$x\in H_{\ast }(K^{n}),y\in H_{\ast }(K^{m}),y=i_{\ast }^{nm}(x),$$ and there is no other $x$ satisfying this condition, then $x$ and $y$ may be thought of as representing the same homology class of the geometric object behind the filtration. In particular, this algebraic approach allows us to resolve the ambiguity: $f_{1,2}(a)= f_{1,2}(b)= c$, is c identified with $a$ or $b$?

The homology group of filtration $\{K^{n}\}$ is defined as the product of the kernels of the inclusions: \begin{equation*} H_{\ast }(\{K^{n}\})=\ker i_{\ast }^{1,2}\oplus \ker \,i_{\ast }^{2,3}\oplus \ldots \oplus \ker i_{\ast }^{s,s+1}. \end{equation*} Here, from each group we take only the elements that are about to die. Since each dies only once, there is no double-counting. Since the sequence ends with $0,$ we know that everyone will die eventually. Hence every homology class appears once and only once.

Proposition. If $i_{\ast}^{n,n+1}$ is an isomorphism for each $n=1,2,...,s-1,$ then $$H_{\ast}(\{K^{n}\})=H_{\ast}(K^{1}).$$
Proposition. If $i_{\ast}^{n,n+1}$ is a monomorphism for each $n=1,2,...,s-1,$ then $$H_{\ast}(\{K^{n}\})=H_{\ast}(K^{s}).$$
Proposition. Suppose $\{K^{n},i^{nm},n,m=1,2,...,s\}$ and $\{L^{n},j^{nm},n,m=1,2,...,s\}$ are filtrations. Then $$H_{\ast}(\{K^{n}\sqcup L^{n}\})=H_{\ast}(\{K^{n} \})\oplus H_{\ast}(\{L^{n}\}).$$
Proposition. Suppose $\{K^{n},i^{nm},n,m=1,2,...,s\}$ and $\{L^{n},j^{nm},n,m=1,2,...,s\}$ are filtrations and $f:K^{s}\rightarrow L^{s}$ is a cell map. Then the homology map of the homology groups of these filtrations $f_{\ast}:H_{\ast }(\{K^{n}\})\rightarrow H_{\ast}(\{L^{n}\})$ is well defined as $$f_{\ast}(x_{1},x_{2},...,x_{s})=(f_{\ast}^{1}(x_{1}),f_{\ast}^{2}(x_{2}),...,f_{\ast}^{s}(x_{s})),$$ where $f^{n}$ is the restriction of $f$ to $K^{n}.$
The stability of the homology group of a filtration follows from the stability of its persistence diagram, i.e., the set of points $$\{(birth,death)\}\subset \mathbf{R}^{2}$$ for the generators of the homology groups of the filtration, plus the diagonal. It is proven that $$d_{B} (D(f),D(g))\leq ||f-g||_{\infty },$$ where $d_{B}$ is the bottle-neck distance between the persistence diagrams $D(f),D(g)$ of two filtrations generated by tame functions $f,g.$ Function $F(x,y)=y-x$ creates an analogue bottle-neck distance for the set of points $\{persistence\}\subset \mathbf{R}$ and its stability follows from the continuity of $F$.