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Difference between revisions of "Klein surface"

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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055510/k0555101.png" /> (see Fig. b) by identifying the points of the line segments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055510/k0555102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055510/k0555103.png" /> lying on the lines parallel to the side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055510/k0555104.png" /> and the points of the segments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055510/k0555105.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055510/k0555106.png" /> symmetric with respect to the centre of the square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055510/k0555107.png" />.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055510/k0555108.png" />-dimensional Euclidean space, but not in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055510/k0555109.png" />-dimensional space.
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055510/k05551010.png" /> is: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055510/k05551011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055510/k05551012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055510/k05551013.png" />. Its Euler characteristic is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055510/k05551014.png" />. Together with the torus it is the only smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055510/k05551015.png" />-dimensional surface which admits deformations of the identity mapping that have no fixed points.
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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=16791
This article was adapted from an original article by E.V. Shikin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article