PoincareDodecahedralSpace 2003-09-05 04:57:00 2007-07-08 18:38:21 Example homology sphere % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Poincar\'e originally conjectured \cite{poincare1900} that a \PMlinkname{homology 3-sphere}{HomologySphere} must be homeomorphic to $S^3$. He later found a counterexample based on the group of rotations of the \PMlinkname{regular dodecahedron}{RegularPolyhedron}, and restated his conjecture in \PMlinkescapetext{terms} of the fundamental group. (See \cite{poincare1904}). To be accurate, the restatement took the form of a question. However it has always been referred to as Poincar\'e's Conjecture.) This conjecture was one of the \PMlinkexternal{Clay Mathematics Institute's}{http://www.claymath.org/} Millennium Problems. It was finally proved by \PMlinkname{Grisha Perelman}{GrigoriPerelman} as a corollary of his \PMlinkescapetext{work} on \PMlinkname{Thurston's geometrization conjecture}{ThurstonsGeometrizationConjecture}. Perelman was awarded the \PMlinkname{Fields Medal}{FieldsMedal} for this work, but he \PMlinkexternal{declined the award}{http://news.bbc.co.uk/2/hi/science/nature/5274040.stm}. Perelman's manuscripts can be found at the arXiv: \cite{perelman2002a}, \cite{perelman2002b}, \cite{perelman2002c}. Here we take a quick look at Poincar\'e's example. Let $\Gamma$ be the rotations of the \PMlinkname{regular dodecahedron}{RegularPolyhedron}. It is easy to check that $\Gamma\cong A_5$. (Indeed, $\Gamma$ \PMlinkname{permutes transitively}{GroupAction} the 6 pairs of \PMlinkescapetext{opposite} faces, and the stabilizer of any pair induces a dihedral group of \PMlinkname{order}{OrderGroup} 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 \PMlinkescapetext{lift} of $\Gamma$ to $SU(2)$ will be denoted $\hat\Gamma$. Hence $P=SU(2)/\hat\Gamma$. $\hat\Gamma$ is a nontrivial central \PMlinkescapetext{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 \PMlinkname{perfect}{PerfectGroup}. Now $\pi_1(P)\cong\hat\Gamma$, whence $H^1(P)=0$ (since it is the abelianization of $\hat\Gamma$). By Poincar\'e duality and the \PMlinkname{universal coefficient theorem}{UniversalCoefficentTheorem}, $H^2(P)\cong0$ as well. Thus, $P$ is indeed a homology sphere. The dodecahedron is a fundamental \PMlinkescapetext{cell} in a tiling of hyperbolic 3-space, and hence $P$ can also be realized by gluing the \PMlinkescapetext{opposite} faces of a \PMlinkescapetext{solid} dodecahedron. Alternatively, Dehn showed how to construct this same example using surgery around a trefoil. Dale Rolfson's fun book \cite{rolfson1976} has more on the surgical view of Poincar\'e's example. \begin{thebibliography}{9900000000} \bibitem{perelman2002a} G. Perelman, \PMlinkexternal{``The entropy formula for the Ricci flow and its geometric applications''}{http://arxiv.org/abs/math.DG/0211159/}, \bibitem{perelman2002b} G. Perelman, \PMlinkexternal{``Ricci flow with surgery on three-manifolds''}{http://arxiv.org/abs/math.DG/0303109/}, \bibitem{perelman2002c} G. Perelman, \PMlinkexternal{``Finite extinction time for the solutions to the Ricci flow on certain three-manifolds''}{http://arxiv.org/abs/math.DG/0307245/}. \bibitem{poincare1900} H. Poincar\'e, ``Second compl\'ement \`a l'analysis situs'', Proceedings of the LMS, 1900. \bibitem{poincare1904} H. Poincar\'e, ``Cinqui\`eme compl\'ement \`a l'analysis situs'', Proceedings of the LMS, 1904. \bibitem{rolfson1976} D. Rolfson, Knots and Links. Publish or Perish Press, 1976. \end{thebibliography}