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. [[ | + | 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 |
$$z_1^2+z_2^3+z_3^5=0$$ | $$z_1^2+z_2^3+z_3^5=0$$ | ||
| − | in $\mathbf C^2$ with the unit sphere. Finally, the icosahedral space can be identified with the [[ | + | in $\mathbf C^2$ with the unit sphere. Finally, the icosahedral space can be identified with the [[dodecahedral space]]. |
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====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> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Seifert, W. Threlfall, "Lehrbuch der Topologie" , Chelsea, reprint (1947)</TD></TR> | ||
| + | </table> | ||
Revision as of 12:18, 10 April 2023
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.
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=53742
Icosahedral space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Icosahedral_space&oldid=53742
This article was adapted from an original article by A.V. Chernavskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article