cellular homology

If $X$ is a cell space, then let $({𝒞}_{*}(X),𝔡)$ be the cell complex where the $n$-th group${𝒞}_n(X)$ is the free abelian group on the cells of dimension $n$, and the boundary mapis as follows: If $e^n$ is an $n$-cell, then we can define a map ${\phi }_f:\partial e^n\to f^{n-1}$, where$f^{n-1}$ is any cell of dimension $n-1$ by the following rule: let $\phi :e^n\to {\mathrm{sk}}_{n-1}X$ be the attaching mapfor $e^n$, where ${\mathrm{sk}}_{n-1}X$ is the $(n-1)$-skeleton of $X$. Then let ${\pi }_f$ be the natural projection

$${\pi }_f:{\mathrm{sk}}_{n-1}X\to {\mathrm{sk}}_{n-1}X/({\mathrm{sk}}_{n-1}X-f)\cong f/\partial f.$$

Let ${\phi }_f={\pi }_f\circ \phi$. Now, $f/\partial f$ is a (n-1)-sphere, so the map ${\phi }_f$ has a degree$\mathrm{deg}f$ which we use to define the boundary operator:

$$𝔡([e^n])=\sum _{dimf=n-1}(\mathrm{deg}{\phi }_f)[f^{n-1}].$$

The resulting chain complex is called the cellular chain complex.

Theorem 1

The homology of the cellular complex is the same as the singular homology of the space. That is

$$H_{*}(𝒞,𝔡)=H_{*}(C,\partial ).$$

Cellular homology is tremendously useful for computations because the groups involved are finitelygenerated.