Difference between revisions of "Klein surface"
From Encyclopedia of Mathematics
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''Klein bottle'' | ''Klein bottle'' | ||
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Figure: k055510a | Figure: k055510a | ||
| − | The Klein surface can be obtained from a square | + | The Klein surface can be obtained from a square $ABCD$ (see Fig. b) by identifying the points of the line segments $AB$ and $CD$ lying on the lines parallel to the side $AD$ and the points of the segments $BC$ and $AD$ symmetric with respect to the centre of the square $ABCD$. |
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/k055510b.gif" /> | <img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/k055510b.gif" /> | ||
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Figure: k055510b | Figure: k055510b | ||
| − | The Klein surface can be topologically imbedded in the | + | The Klein surface can be topologically imbedded in the $4$-dimensional Euclidean space, but not in $3$-dimensional space. |
Attention was drawn to the Klein surface by F. Klein (1874). | Attention was drawn to the Klein surface by F. Klein (1874). | ||
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====Comments==== | ====Comments==== | ||
| − | A Klein bottle can be obtained by glueing together two crosscaps (Möbius bands, cf. [[Möbius strip|Möbius strip]]) along their boundaries. The homology of the Klein bottle | + | A Klein bottle can be obtained by glueing together two crosscaps (Möbius bands, cf. [[Möbius strip|Möbius strip]]) along their boundaries. The homology of the Klein bottle $K$ is: $H_0(K;\mathbf Z)=\mathbf Z$, $H_1(K;\mathbf Z)=\mathbf Z\oplus\mathbf Z/(2)$, $H_2(K;\mathbf Z)=0$. Its Euler characteristic is $\chi(K)=0$. Together with the torus it is the only smooth $2$-dimensional surface which admits deformations of the identity mapping that have no fixed points. |
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D.W. Blackett, "Elementary topology" , Acad. Press (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> K. Jänich, "Topology" , Springer (1984) (Translated from German)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Mayer, "Algebraic topology" , Prentice-Hall (1972)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D.W. Blackett, "Elementary topology" , Acad. Press (1967)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> K. Jänich, "Topology" , Springer (1984) (Translated from German)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J. Mayer, "Algebraic topology" , Prentice-Hall (1972)</TD></TR></table> | ||
Latest revision as of 19:36, 17 April 2014
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 $ABCD$ (see Fig. b) by identifying the points of the line segments $AB$ and $CD$ lying on the lines parallel to the side $AD$ and the points of the segments $BC$ and $AD$ symmetric with respect to the centre of the square $ABCD$.
Figure: k055510b
The Klein surface can be topologically imbedded in the $4$-dimensional Euclidean space, but not in $3$-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 $K$ is: $H_0(K;\mathbf Z)=\mathbf Z$, $H_1(K;\mathbf Z)=\mathbf Z\oplus\mathbf Z/(2)$, $H_2(K;\mathbf Z)=0$. Its Euler characteristic is $\chi(K)=0$. Together with the torus it is the only smooth $2$-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) |
How to Cite This Entry:
Klein surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Klein_surface&oldid=31839
Klein surface. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Klein_surface&oldid=31839
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article