Poincaré dodecahedral space

Poincaré originally conjectured [4] that a homologyPlanetmathPlanetmath 3-sphere (http://planetmath.org/HomologySphere) must be homeomorphic to S3. 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 groupMathworldPlanetmathPlanetmath. (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 Γ be the rotations of the regular dodecahedron (http://planetmath.org/RegularPolyhedron). It is easy to check that ΓA5. (Indeed, Γ permutes transitively (http://planetmath.org/GroupAction) the 6 pairs of faces, and the stabilizerMathworldPlanetmath of any pair induces a dihedral groupMathworldPlanetmath of order (http://planetmath.org/OrderGroup) 10.) In particular, Γ is perfectPlanetmathPlanetmath. Let P be the quotient spaceMathworldPlanetmath P=SO3()/Γ. We check that P is a homology sphere.

To do this it is easier to work in the universal coverMathworldPlanetmath SU(2) of SO3(), since SU(2)S3. The of Γ to SU(2) will be denoted Γ^. Hence P=SU(2)/Γ^. Γ^ is a nontrivial central of A5 by {±I}, which means that there is no splitting to the surjection Γ^Γ. In fact A5 has no nonidentity 2-dimensional unitary representationsMathworldPlanetmath. In particular, Γ^, like Γ, is perfect (http://planetmath.org/PerfectGroup).

Now π1(P)Γ^, whence H1(P)=0 (since it is the abelianizationMathworldPlanetmath of Γ^). By Poincaré duality and the universal coefficient theorem (http://planetmath.org/UniversalCoefficentTheorem), H2(P)0 as well. Thus, P is indeed a homology sphere.

The dodecahedronMathworldPlanetmath 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.


  • 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