# Complex projective plane

This article is about a particular topological space (uniquely determined up to homeomorphism)|View a complete list of particular topological spaces

## Contents

## Definition

The **complex projective plane** is the complex projective space of complex dimension 2. As a manifold over the reals, it has dimension 4. It is denoted or .

Alternatively, it can be viewed as the quotient of the space under the action of by multiplication. In particular, there is a fibration .

## Algebraic topology

### Homology groups

`Further information: homology of complex projective space`

The homology groups with coefficients in are as follows: , and all other homology groups are zero.

More generally, the homology group with coefficients in a module over a commutative unital ring are as follows: , and all other homology groups are zero.

### Cohomology groups

`Further information: cohomology of complex projective space`

The cohomology groups with coefficients in are as follows: , and all other cohomology groups are zero. The cohomology ring is where is an additive generator for the second cohomology group.

More generally, the cohomology group with coefficients in a commutative unital ring are as follows: , and all other cohomology groups are zero. The cohomology ring is where is a -module generator for the second cohomology module.

### Homotopy groups

`Further information: homotopy of complex projective space`

The homotopy groups are as follows:

Value of | General name for homotopy group/set | What is for generic ?) | What is ? |
---|---|---|---|

0 | set of path components | one-point set | one-point set, so is a path-connected space |

1 | fundamental group | trivial group | trivial group, so is a simply connected space. |

2 | second homotopy group | ||

3 | third homotopy group | trivial group | trivial group |

4 | fourth homotopy group | trivial group | trivial group |

5 | fifth homotopy group | if , zero otherwise | |

Same as |