Homology group of filtration

we combine the homology groups of all the cell complexes into one structure. For example, we may have a sequence of complexes: \[ K^{1}\hookrightarrow K^{2}\hookrightarrow K^{3}\hookrightarrow K^{4}% \hookrightarrow\ldots\ \hookrightarrow K^{s}, \] where the arrows are inclusions: $i^{n,n+1}:K^{n}\hookrightarrow K^{n+1},$ and let $i^{n,m}:K^{n}\hookrightarrow K^{m}$ be defined as the compositions. This structure $\{K^{n}\}$ is called a filtration. Further, each of these inclusions generates a homology homomorphism: $i_{\ast}% ^{n,m}:H_{\ast}(K^{n})\hookrightarrow H_{\ast}(K^{m}).$ Commonly, these are simply linear operators represented by matrices. Then we have a sequence of homology groups connected by homomorphisms: \[ H_{\ast}(K^{1})\rightarrow H_{\ast}(K^{2})\rightarrow\ldots\ \rightarrow H_{\ast}(K^{s})\longrightarrow0. \] The homology group of filtration $\{K^{n}\}$ captures all homology classes of all the complexes in a compact way: \[ H_{\ast}(\{K^{n}\})=\ker i_{\ast}^{1,2}\oplus\ker\,i_{\ast}^{2,3}\oplus\ker i_{\ast}^{3,4}\oplus\ldots\oplus\ker i_{\ast}^{s,s+1}. \] Indeed, from the homology group of each complex we take only the elements that are about to die (go to $0$). Since each dies only once, there is no double counting.