Klein bottle: Difference between revisions

Latest revision as of 20:10, 2 April 2011

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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 $H_{0}=\mathbb {Z}$ , $H_{1}=\mathbb {Z} \oplus \mathbb {Z} /2\mathbb {Z}$ , $H_{2}=0$ , 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 $0$ .

Cohomology

Further information: cohomology of Klein bottle

Homotopy

Further information: homotopy of Klein bottle

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 {{further|[[homology of Klein bottle]]}}
+The Klein bottle has <math>H_0 = \mathbb{Z}</math>, <math>H_1 = \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}</math>, <math>H_2 = 0</math>, and all higher homology groups are zero. The [[Betti number]]s are <math>b_0 = b_1 = 1</math>, higher <math>b_k</math>s are zero, the [[Poincare polynomial]] is <math>1 + x</math>, and the [[Euler characteristic]] is thus <math>0</math>.
 ===Cohomology===