Homology defined on CW complexes.

Idea:

n-chain ~ $\mathbb{Z}\langle \{n-cells\}\rangle$

boundary ~ degree of the attaching map on each cell.

For $X$ a CW complex, we define the cellular homolgy group as

$H^{CW}_n X := H_n( ... \to H_{n+1}(X^{n+1},X^n)$ $\to_{d_{n+1}} H_n(X^n,X^{n-1}) \to_{d_n} H_{n-1}(X^{n-1},X^{n-2})\to...)$

where $X^i$ is the $i$-skeleton of the CW complex $X$. $H_n(A,B)$ is the Relative homology group of space A relative to B. The outermost $H_n$ means the $n$th Homology group of the Chain complex it acts on. This chain complex is formed by the relative homology groups together with the maps $d_n$.

The map $d_n$ is defined as the composition $j_{n-1} \circ \partial_n$ where these are the maps in the sequence $...\to H_n(X^n,X^{n-1}) \to_{\partial_n} H_{n-1}(X^{n-1})\to_{j_{n-1}} H_{n-1}(X^{n-1},X^{n-2})\to ...$ ( you can see the composition has the right domain and codomain). This sequence comes from putting together a few LES of the skeletons as follows:

The middle chain is called the Cellular chain complex. See here

Cellular boundary formula

We can write a formula for $d_n$. See here