# Homology of Klein bottle

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 and the topological space/family is Klein bottle

Get more specific information about Klein bottle | Get more computations of homology

## Statement

The homology of the Klein bottle is given by:

and:

All other homology groups are 0.

## Related invariants

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

Invariant | General description | Description of value for Klein bottle |
---|---|---|

Betti numbers | The Betti number is the rank of the homology group. | , for |

Poincare polynomial | Generating polynomial for Betti numbers | |

Euler characteristic | 0 -- consistent with the fact that its double cover is the 2-torus, which, being a nontrivial compact connected Lie group, has Euler characteristic zero, and the fact that Euler characteristic of covering space is product of degree of covering and Euler characteristic of base |