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Difference between revisions of "Icosahedral space"

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The three-dimensional space that is the orbit space of the action of the binary icosahedron group on the three-dimensional sphere. It was discovered by H. Poincaré as an example of a homology sphere of genus 2 in the consideration of Heegaard diagrams (cf. [[Heegaard diagram|Heegaard diagram]]). The icosahedral space is a $p$-sheeted covering of $S^3$ ramified along a torus knot of type $(q,r)$, where $p,q,r$ is any permutation of the numbers $2,3,5$. The icosahedral space can be defined analytically as the intersection of the surface
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The three-dimensional space that is the orbit space of the action of the [[binary icosahedral group]] on the three-dimensional sphere. It was discovered by H. Poincaré as an example of a homology sphere of genus 2 in the consideration of Heegaard diagrams (cf. [[Heegaard diagram]]). The icosahedral space is a $p$-sheeted covering of $S^3$ ramified along a torus knot of type $(q,r)$, where $p,q,r$ is any permutation of the numbers $2,3,5$. The icosahedral space can be defined analytically as the intersection of the surface
  
 
$$z_1^2+z_2^3+z_3^5=0$$
 
$$z_1^2+z_2^3+z_3^5=0$$
  
in $\mathbf C^2$ with the unit sphere. Finally, the icosahedral space can be identified with the [[Dodecahedral space|dodecahedral space]].
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in $\mathbf C^2$ with the unit sphere. Finally, the icosahedral space can be identified with the [[dodecahedral space]].
 
 
 
 
 
 
====Comments====
 
 
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Seifert,  W. Threlfall,  "Lehrbuch der Topologie" , Chelsea, reprint  (1947)</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Seifert,  W. Threlfall,  "Lehrbuch der Topologie" , Chelsea, reprint  (1947)</TD></TR>
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</table>

Latest revision as of 12:19, 10 April 2023

The three-dimensional space that is the orbit space of the action of the binary icosahedral group on the three-dimensional sphere. It was discovered by H. Poincaré as an example of a homology sphere of genus 2 in the consideration of Heegaard diagrams (cf. Heegaard diagram). The icosahedral space is a $p$-sheeted covering of $S^3$ ramified along a torus knot of type $(q,r)$, where $p,q,r$ is any permutation of the numbers $2,3,5$. The icosahedral space can be defined analytically as the intersection of the surface

$$z_1^2+z_2^3+z_3^5=0$$

in $\mathbf C^2$ with the unit sphere. Finally, the icosahedral space can be identified with the dodecahedral space.

References

[a1] H. Seifert, W. Threlfall, "Lehrbuch der Topologie" , Chelsea, reprint (1947)
How to Cite This Entry:
Icosahedral space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Icosahedral_space&oldid=32908
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article