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 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. Further information: composite of consecutive maps of cellular chain complex is zero

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 $X$. Further information: cellular homology equals singular homology

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

Functoriality

Further information: Cellular chain complex functor

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.