# cellular homology

If $X$ is a cell space, then let $({\mathcal{C}}_{*}(X),\U0001d521)$ 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 ${\phi}_{f}:\partial {e}^{n}\to {f}^{n-1}$, 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 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 ${\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:

$$\U0001d521([{e}^{n}])=\sum _{dimf=n-1}(\mathrm{deg}{\phi}_{f})[{f}^{n-1}].$$ |

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

$${H}_{*}(\mathcal{C},\U0001d521)={H}_{*}(C,\partial ).$$ |

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

Title | cellular homology |
---|---|

Canonical name | CellularHomology |

Date of creation | 2013-03-22 13:14:55 |

Last modified on | 2013-03-22 13:14:55 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 6 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 55N10 |