# Klein bottle

From Topospaces

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## Definition

The **Klein bottle** is a compact non-orientable surface (and hence, in particular, a connected two-dimensional manifold) defined in the following equivalent ways (up to homeomorphism)

- It is the connected sum of two copies of the real projective plane.
- It is obtained by taking a torus, removing one of the factor circles, and re-gluing this circle with the opposite orientation.

(More definitions, more precise definitions needed).

The Klein bottle is one of the compact non-orientable surfaces.

## Algebraic topology

### Homology

`Further information: homology of Klein bottle`

The Klein bottle has , , , and all higher homology groups are zero. The Betti numbers are , higher s are zero, the Poincare polynomial is , and the Euler characteristic is thus .

### Cohomology

`Further information: cohomology of Klein bottle`

### Homotopy

`Further information: homotopy of Klein bottle`