If $X$ is a cell space, then let $(\mc{C}_*(X),\fr{d})$ be the cell complex where the $n$ -th group $\mc{C}_n(X)$ is the free abelian group on the cells of dimension $n$ , and the boundary map is as follows: If $e^n$ is an $n$ -cell, then we can define a map $\vp_f:\partial e^n\to f^{n-1}$ , where $f^{n-1}$ is any cell of dimension $n-1$ by the following rule: let $\vp:e^n\to\mathrm{sk}_{n-1} X$ be the attaching map for $e^n$ , where $\sk_{n-1}X$ is the $(n-1)$ -skeleton of $X$ . Then let $\pi_f$ be the natural projection $$\pi_f:\mathrm{sk}_{n-1}X\to \mathrm{sk}_{n-1} X/(\mathrm{sk}_{n-1} X-f)\cong f/\partial f.$$ Let $\vp_f=\pi_f\circ\vp$ . Now, $f/\partial f$ is a (n-1)-sphere, so the map $\vp_f$ has a degree $\deg f$ which we use to define the boundary operator: $$\fr{d}([e^n])=\sum_{\dim f=n-1}(\deg \vp_f)[f^{n-1}].$$

The resulting chain complex is called the cellular chain complex.

Cellular homology is tremendously useful for computations because the groups involved are finitely generated.