# Homology of complex projective space

From Topospaces

This article describes the value (and the process used to compute it) of some homotopy invariant(s) for a topological space or family of topological spaces. The invariant is homology group and the topological space/family is complex projective space

Get more specific information about complex projective space | Get more computations of homology group

## Contents

## Statement

### Unreduced version with coefficients in integers

### Reduced version with coefficients in integers

### Unreduced version with coefficients in an abelian group or module

For coefficients in an abelian group , the homology groups are:

## Homology groups with integer coefficients in tabular form

We illustrate how the homology groups work for small values of (whereby the dimension of the corresponding complex projective space is ). Note that for , all homology groups are zero, so we omit those cells for visual clarity.

Complex projective space | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 2-sphere | 0 | ||||||||

2 | complex projective plane | 0 | 0 | |||||||

3 | CP^3 | 0 | 0 | 0 | ||||||

4 | CP^4 | 0 | 0 | 0 | 0 |

## Related invariants

These are all invariants that can be computed in terms of the homology groups.

Invariant | General description | Description of value for complex projective space |
---|---|---|

Betti numbers | The Betti number is the rank of the torsion-free part of the homology group. | , all other values are zero. |

Poincare polynomial | Generating polynomial for Betti numbers | |

Euler characteristic |

## Facts used

## Proof

We use the CW-complex structure on complex projective space that has exactly one cell in every even dimension till . The cellular chain complex of this thus has s in all the even positions till , and hence its homology is in all even dimensions till .