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 relative homology group $$H_n(X^n, X^{n-1})$$
 * The boundary map is defined as follows. First note that the defining ingredient::long exact sequence of homology of a pair $$(X^n,X^{n-1})$$ gives a map:

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

The long exact sequence of homology of a pair $$(X^{n-1},X^{n-2})$$ gives a map:

$$H_{n-1}(X^{n-1}) \to H_{n-1}(X^{n-1},X^{n-2})$$.

Composing these two maps, we get the boundary map for the chain complex:

$$H_n(X^n,X^{n-1}) \to H_{n-1}(X^{n-1},X^{n-2})$$

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 $$n^{th}$$ singular homology of the pair $$(X,X^{-1})$$ ($$X^{-1}$$ can be viewed as the base space). In particular, if $$X^{-1}$$ is empty, i.e., the filtration begins with the empty set, then the cellular homology of the filtration equals the singular homology of $$X$$.

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

Functoriality
The cellular chain complex can be viewed as a functor from the category of cellular spaces with cellular maps, to the category of chain complexes with chain maps.