# cellular homology

If $X$ is a cell space, then let $(\mathcal{C}_{*}(X),\mathfrak{d})$ be the cell complex where the $n$-th group $\mathcal{C}_{n}(X)$ is the free abelian group on the cells of dimension $n$, and the boundary map is as follows: If $e^{n}$ is an $n$-cell, then we can define a map $\varphi_{f}:\partial e^{n}\to f^{n-1}$, where $f^{n-1}$ is any cell of dimension $n-1$ by the following rule: let $\varphi:e^{n}\to\mathrm{sk}_{n-1}X$ be the attaching map for $e^{n}$, 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 $\varphi_{f}=\pi_{f}\circ\varphi$. Now, $f/\partial f$ is a (n-1)-sphere, so the map $\varphi_{f}$ has a degree $\deg f$ which we use to define the boundary operator:

 $\mathfrak{d}([e^{n}])=\sum_{\dim f=n-1}(\deg\varphi_{f})[f^{n-1}].$

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

 $H_{*}(\mathcal{C},\mathfrak{d})=H_{*}(C,\partial).$

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

Title cellular homology CellularHomology 2013-03-22 13:14:55 2013-03-22 13:14:55 bwebste (988) bwebste (988) 6 bwebste (988) Definition msc 55N10