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 $n^{th}$ 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 $n^{th}$ homology group of the cellular chain complex, is isomorphic to the $n^{th}$ 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.