Poincaré dodecahedral space

Poincaré originally conjectured [4] that a homology 3-sphere (http://planetmath.org/HomologySphere) must be homeomorphic to $S^3$. He later found a counterexample based on the group of rotations of theregular dodecahedron (http://planetmath.org/RegularPolyhedron), and restated his conjecture in of the fundamental group. (See [5]). To be accurate, the restatement took the form of a question. However it has always been referred to as Poincaré’s Conjecture.)

This conjecture was one of the http://www.claymath.org/Clay Mathematics Institute’s Millennium Problems. It was finally proved by Grisha Perelman (http://planetmath.org/GrigoriPerelman) as acorollary of his on Thurston’s geometrization conjecture (http://planetmath.org/ThurstonsGeometrizationConjecture). Perelman was awarded the Fields Medal (http://planetmath.org/FieldsMedal) for this work, but hehttp://news.bbc.co.uk/2/hi/science/nature/5274040.stmdeclined the award.Perelman’s manuscripts can be found at the arXiv: [1], [2],[3].

Here we take a quick look at Poincaré’s example. Let $\mathrm{\Gamma }$ be the rotations of theregular dodecahedron (http://planetmath.org/RegularPolyhedron). It is easy to check that $\mathrm{\Gamma }\cong A_5$. (Indeed, $\mathrm{\Gamma }$ permutes transitively (http://planetmath.org/GroupAction) the 6 pairs of faces, and the stabilizer of any pair induces a dihedral group of order (http://planetmath.org/OrderGroup) 10.) In particular, $\mathrm{\Gamma }$ is perfect. Let $P$ be the quotient space $P=S{O}_3(ℝ)/\mathrm{\Gamma }$. We check that $P$ is a homology sphere.

To do this it is easier to work in the universal cover $SU(2)$ of $S{O}_3(ℝ)$, since $SU(2)\cong S^3$. The of $\mathrm{\Gamma }$ to $SU(2)$ will be denoted $\widehat{\mathrm{\Gamma }}$. Hence $P=SU(2)/\widehat{\mathrm{\Gamma }}$. $\widehat{\mathrm{\Gamma }}$ is a nontrivial central of $A_5$ by $\{\pm I\}$, which means that there is no splitting to the surjection $\widehat{\mathrm{\Gamma }}\to \mathrm{\Gamma }$. In fact $A_5$ has no nonidentity 2-dimensional unitary representations. In particular, $\widehat{\mathrm{\Gamma }}$, like $\mathrm{\Gamma }$, is perfect (http://planetmath.org/PerfectGroup).

Now ${\pi }_1(P)\cong \widehat{\mathrm{\Gamma }}$, whence $H^1(P)=0$ (since it is the abelianization of $\widehat{\mathrm{\Gamma }}$). By Poincaré duality and the universal coefficient theorem (http://planetmath.org/UniversalCoefficentTheorem), $H^2(P)\cong 0$ as well. Thus, $P$ is indeed a homology sphere.

The dodecahedron is a fundamental in a tiling of hyperbolic 3-space, and hence $P$ can also be realized by gluing the faces of a dodecahedron. Alternatively, Dehn showed how to construct this same example using surgery around a trefoil.Dale Rolfson’s fun book [6] has more on the surgical view of Poincaré’s example.