Poincaré originally conjectured [4] that a homology 3-sphere must be homeomorphic to . He later found a counterexample based on the group of rotations of the regular dodecahedron, and restated his conjecture in terms 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 Clay Mathematics Institute's Millennium Problems. It was finally proved by Grisha Perelman as a corollary of his work on Thurston's geometrization conjecture. Perelman was awarded the Fields Medal for this work, but he declined 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 be the rotations of the regular dodecahedron. It is easy to check that . (Indeed, permutes transitively the 6 pairs of opposite faces, and the stabilizer of any pair induces a dihedral group of order 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 lift of to will be denoted . Hence . is a nontrivial central extension 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.

Now , whence (since it is the abelianization of ). By Poincaré duality and the universal coefficient theorem, as well. Thus, is indeed a homology sphere.

The dodecahedron is a fundamental cell in a tiling of hyperbolic 3-space, and hence can also be realized by gluing the opposite faces of a solid 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.

Bibliography

1

G. Perelman, ``The entropy formula for the Ricci flow and its geometric applications'',

2

G. Perelman, ``Ricci flow with surgery on three-manifolds'',

3

G. Perelman, ``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.