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 the regular 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 a corollary 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 he http://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 $\Gamma$ be the rotations of the regular dodecahedron (http://planetmath.org/RegularPolyhedron). It is easy to check that $\Gamma\cong A_{5}$ . (Indeed, $\Gamma$ 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, $\Gamma$ is perfect. Let $P$ be the quotient space $P=SO_{3}(\mathbb{R})/\Gamma$ . We check that $P$ is a homology sphere.

To do this it is easier to work in the universal cover $SU(2)$ of $SO_{3}(\mathbb{R})$ , since $SU(2)\cong S^{3}$ . The of $\Gamma$ to $SU(2)$ will be denoted $\hat{\Gamma}$ . Hence $P=SU(2)/\hat{\Gamma}$ . $\hat{\Gamma}$ is a nontrivial central of $A_{5}$ by $\{\pm I\}$ , which means that there is no splitting to the surjection $\hat{\Gamma}\to\Gamma$ . In fact $A_{5}$ has no nonidentity 2-dimensional unitary representations. In particular, $\hat{\Gamma}$ , like $\Gamma$ , is perfect (http://planetmath.org/PerfectGroup).

Now $\pi_{1}(P)\cong\hat{\Gamma}$ , whence $H^{1}(P)=0$ (since it is the abelianization of $\hat{\Gamma}$ ). 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.

References

1 G. Perelman, http://arxiv.org/abs/math.DG/0211159/“The entropy formula for the Ricci flow and its geometric applications”,
2 G. Perelman, http://arxiv.org/abs/math.DG/0303109/“Ricci flow with surgery on three-manifolds”,
3 G. Perelman, http://arxiv.org/abs/math.DG/0307245/“Finite extinction time for the solutions to the Ricci flow on certain three-manifolds”.
4 H. Poincaré, “Second complément à l’analysis situs”, Proceedings of the LMS, 1900.
5 H. Poincaré, “Cinquième complément à l’analysis situs”, Proceedings of the LMS, 1904.
6 D. Rolfson, Knots and Links. Publish or Perish Press, 1976.

Title	Poincaré dodecahedral space
Canonical name	PoincareDodecahedralSpace
Date of creation	2013-03-22 13:56:21
Last modified on	2013-03-22 13:56:21
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	65
Author	Mathprof (13753)
Entry type	Example
Classification	msc 57R60