Cellular chain complex

Definition

The cellular chain complex of a cellular space $X$ (viz, a topological space $X$ equipped with a cellular filtration $X^{n}$ ) is described as follows:

The $n^{t h}$ member is the group $H_{n} (X^{n}, X^{n - 1})$
The boundary map is defined as follows. First note that the long exact sequence of homology of a pair $X^{n}, X^{n - 1})$ gives a map:

$H_{n} (X^{n}, X^{n - 1}) \to H_{n - 1} (X^{n - 1})$

We compose this with the natural map from $H_{n - 1} (X^{n - 1})$ to $H_{n - 1} (X^{n - 1}, X^{n - 2})$ .

The fact that the composite of two boundary maps is zero, follows from the trick of writing each chain map as a composite of the two maps as above, and then noting that in the composite, we get a composite of two consecutive terms of a long exact sequence of homology.

Facts

The $n^{t h}$ homology group of the cellular chain complex, is isomorphic to the $n^{t h}$ homology of the pair $(X, X^{- 1})$ ( $X^{- 1}$ can be viewed as the base space).

Cellular homology is typically used only for cellular filtrations arising from CW complex structures.