Klein surface

Klein bottle

A closed one-sided surface of genus 1 (see Fig. a, a, b).

Figure: k055510a

The Klein surface can be obtained from a square (see Fig. b) by identifying the points of the line segments and lying on the lines parallel to the side and the points of the segments and symmetric with respect to the centre of the square .

Figure: k055510b

The Klein surface can be topologically imbedded in the -dimensional Euclidean space, but not in -dimensional space.

Attention was drawn to the Klein surface by F. Klein (1874).

Comments

A Klein bottle can be obtained by glueing together two crosscaps (Möbius bands, cf. Möbius strip) along their boundaries. The homology of the Klein bottle is: , , . Its Euler characteristic is . Together with the torus it is the only smooth -dimensional surface which admits deformations of the identity mapping that have no fixed points.

References

[a1]	D.W. Blackett, "Elementary topology" , Acad. Press (1967)
[a2]	K. Jänich, "Topology" , Springer (1984) (Translated from German)
[a3]	J. Mayer, "Algebraic topology" , Prentice-Hall (1972)