Difference between revisions of "Icosahedral space"
From Encyclopedia of Mathematics
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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 | + | {{TEX|done}} |
| + | 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 | ||
| − | + | $$z_1^2+z_2^3+z_3^5=0$$ | |
| − | in | + | in $\mathbf C^2$ with the unit sphere. Finally, the icosahedral space can be identified with the [[Dodecahedral space|dodecahedral space]]. |
Revision as of 12:15, 14 August 2014
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). 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.
Comments
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
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