Poincaré dodecahedral space
Poincaré originally conjectured  that a homology 3-sphere (http://planetmath.org/HomologySphere) must be homeomorphic to . 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 ). 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: , , .
Here we take a quick look at Poincaré’s example. Let be the rotations of the regular dodecahedron (http://planetmath.org/RegularPolyhedron). It is easy to check that . (Indeed, 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, is perfect. Let be the quotient space . We check that is a homology sphere.
To do this it is easier to work in the universal cover of , since . The of to will be denoted . Hence . is a nontrivial central of by , which means that there is no splitting to the surjection . In fact has no nonidentity 2-dimensional unitary representations. In particular, , like , is perfect (http://planetmath.org/PerfectGroup).
Now , whence (since it is the abelianization of ). By Poincaré duality and the universal coefficient theorem (http://planetmath.org/UniversalCoefficentTheorem), as well. Thus, is indeed a homology sphere.
The dodecahedron is a fundamental in a tiling of hyperbolic 3-space, and hence 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  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|
|Date of creation||2013-03-22 13:56:21|
|Last modified on||2013-03-22 13:56:21|
|Last modified by||Mathprof (13753)|