Poincaré originally conjectured [4] that a homology 3-sphere must be homeomorphic to $S^3$ 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 $\Gamma$ be the rotations of the regular dodecahedron. It is easy to check that $\Gamma\cong A_5$ (Indeed, $\Gamma$ permutes transitively the 6 pairs of opposite faces, and the stabilizer of any pair induces a dihedral group of order 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 lift of $\Gamma$ to $SU(2)$ will be denoted $\hat\Gamma$ Hence $P=SU(2)/\hat\Gamma$ $\hat\Gamma$ is a nontrivial central extension 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.

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, $H^2(P)\cong0$ as well. Thus, $P$ is indeed a homology sphere.

The dodecahedron is a fundamental cell in a tiling of hyperbolic 3-space, and hence $P$ 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.