Icosahedral space

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)