View source for Icosahedral space

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050010/i0500101.png" />-sheeted covering of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050010/i0500102.png" /> ramified along a torus knot of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050010/i0500103.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050010/i0500104.png" /> is any permutation of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050010/i0500105.png" />. The icosahedral space can be defined analytically as the intersection of the surface

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050010/i0500106.png" /></td> </tr></table>

in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050010/i0500107.png" /> with the unit sphere. Finally, the icosahedral space can be identified with the [[Dodecahedral space|dodecahedral space]].



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Seifert,   W. Threlfall,   "Lehrbuch der Topologie" , Chelsea, reprint  (1947)</TD></TR></table>