# 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 $\mathrm{\Gamma}$ be the rotations of the
regular 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^{} $P=S{O}_{3}(\mathbb{R})/\mathrm{\Gamma}$. We check that $P$ is a homology sphere.

To do this it is easier to work in the universal cover^{} $SU(2)$ of $S{O}_{3}(\mathbb{R})$, 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.

## 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 |