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# Poincaré dodecahedral space

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, $H^{2}(P)\cong 0$ 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.

# References

- 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.

## Mathematics Subject Classification

57R60*no label found*

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