Cellular homology

Remarks

The group $H_n(X^n, X^{n-1})$ is free abelian with one generator $e_\alpha$ for each $n$-cell $e_\alpha$ of $X$. In this basis, $d_n(e_\alpha) = \sum_\beta d_{\alpha\beta} e_\beta$, where $d_{\alpha\beta}$ is the degree of the map $S^{n-1} \to X^{n-1} \to S^{n-1}$ given by the attaching map of $e_\alpha$ followed by the collapse of everything outside $e_\beta$. The basic theorem is that $H_n^{\mathrm{CW}}(X) \cong H_n(X)$ for all $n$.

Example: the real projective plane $\R P^2$ has one cell in each of the dimensions $0$, $1$, $2$. The $2$-cell is attached by a map of degree $2$ onto the $1$-skeleton $S^1$, so $d_2 = 2$, and $d_1 = 0$ because the $1$-cell is a loop. The cellular chain complex is $\Z \xrightarrow{2} \Z \xrightarrow{0} \Z$, which gives $H_0 = \Z$, $H_1 = \Z/2$ and $H_2 = 0$.

Non-example: the definition needs a CW structure, that is, a filtration by skeleta in which each $X^n$ comes from $X^{n-1}$ by attaching $n$-cells. For an arbitrary increasing chain of subspaces, the groups $H_n(X^n, X^{n-1})$ need not vanish outside degree $n$, and the homology of the resulting complex need not be the homology of $X$.