Klein bottle

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)


 * 1) It is the connected sum of two copies of the defining ingredient::real projective plane.
 * 2) 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.

Homology
The Klein bottle has $$H_0 = \mathbb{Z}$$, $$H_1 = \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$$, $$H_2 = 0$$, and all higher homology groups are zero. The Betti numbers are $$b_0 = b_1 = 1$$, higher $$b_k$$s are zero, the Poincare polynomial is $$1 + x$$, and the Euler characteristic is thus $$0$$.