GeometrizationConjectureAndTheorem 2009-02-19 13:39:31 2009-02-19 15:46:43 Theorem Haken manifold orientable Haken manifolds Haken manifold orientable Haken manifolds geometrization conjecture and theorem % this is the default PlanetPhysics 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{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym, enumerate} \usepackage{xypic, xspace} \usepackage[mathscr]{eucal} \usepackage[dvips]{graphicx} \usepackage[curve]{xy} % define commands here \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote}}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\grpL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\rO}{{\rm O}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\SL}{{\rm Sl}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\Symb}{{\rm Symb}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} %\newcommand{\grp}{\mathcal G} \renewcommand{\H}{\mathcal H} \renewcommand{\cL}{\mathcal L} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathsf{G}}} \newcommand{\dgrp}{{\mathsf{D}}} \newcommand{\desp}{{\mathsf{D}^{\rm{es}}}} \newcommand{\grpeod}{{\rm Geod}} %\newcommand{\grpeod}{{\rm geod}} \newcommand{\hgr}{{\mathsf{H}}} \newcommand{\mgr}{{\mathsf{M}}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathsf{G)}}} \newcommand{\obgp}{{\rm Ob(\mathsf{G}')}} \newcommand{\obh}{{\rm Ob(\mathsf{H})}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\grphomotop}{{\rho_2^{\square}}} \newcommand{\grpcalp}{{\mathsf{G}(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\grplob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\grpa}{\grpamma} %\newcommand{\grpa}{\grpamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\ovset}[1]{\overset {#1}{\ra}} \newcommand{\ovsetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \newcommand{\<}{{\langle}} \def\baselinestretch{1.1} \hyphenation{prod-ucts} \newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}& #3 \dto^{#5} \\ #6 \rto_{#7} & #8 \enddiagram \eqno{\mbox{#9}}$$ } \def\C{C^{\ast}} \newcommand{\labto}[1]{\stackrel{#1}{\longrightarrow}} \newcommand{\quadr}[4] {\begin{pmatrix} & #1& \\[-1.1ex] #2 & & #3\\[-1.1ex]& #4& \end{pmatrix}} \def\D{\mathsf{D}} \subsection{Thurston's Geometrization Theorem and Conjecture} \subsubsection{History} Haken manifolds are named after Wolfgang Haken (b.1928) because their definition involves incompressible surfaces, and Wolfgang Haken pioneered the use of incompressible surfaces in topology. At the University of Illinois at Urbana-Champaign, Haken and his colleague Kenneth Appel, solved in 1976 one of the most famous problems in mathematics by proving the {\bf four-color theorem}. He also proved that Haken manifolds have a hierarchy. The hierarchy facilitates the proof of several theorems about about Haken manifolds through logical/mathematical induction: first, one proves a theorem for 3-balls, then one proves that if a theorem is true for pieces resulting by ``cutting'' a Haken manifold, that it is true for the whole Haken manifold; it is essential that such ``cutting'' is selected along a surface that was incompressible. This makes possible the proof by induction in most cases by proceeding from one induction step to the next. Thus, Haken showed that there is a finite procedure to find an incompressible surface if the 3-manifold had one; 20 years later, Jaco and Oertel showed again by utilizing induction that there was an algorithm to determine if a 3-manifold was Haken. Because according to Jaco and Oertel there exists an algorithm to find out if a 3-manifold is Haken, the fundamental topological problem of recognizing 3-manifolds can be considered to be solved for the category of Haken manifolds. Furthermore, Friedhelm Waldhausen proved that \emph{closed Haken manifolds} are topologically rigid, and thus such 3-manifolds are completely determined by their fundamental group. One can also conjecture that,in general, CW--complexes $C_{wh}$ of closed Haken manifolds, $M_{cH}$, are completely determined by the fundamental groupoid functor $\F_G$ associated with the category of $C_{wh}$ and CW-homeomorphisms. \subsection{Thurston's Geometrization Theorem} Thurston reported his proof of the geometrization theorem in 1980, and several complete proofs have been published since. Furthermore, in 2003, Grigori (Grisha) Perelman sketched a proof of the general, full geometrization conjecture using Ricci flows with surgery; his proof of the full geometrization conjecture --as reported by specialized mathematicians in 2008-- is said to be essentially correct. Let us consider first a \emph{Haken manifold} which is defined as a compact, $P^2$--irreducible $3$--manifold that contains a two--sided incompressible 2D-surface. (One also considers in topology \emph{orientable Haken manifolds}, in which case the Haken manifold is a compact, orientable and irreducible $3$--manifold that contains an orientable, incompressible surface). Thurston's geometrization theorem, also called ``the Hyperbolization Theorem'', is stated as follows: \begin{theorem} {\bf Thurston Geometrization Theorem (1980):} Haken manifolds satisfy can be decomposed into submanifolds that have geometric structures. \end{theorem} In essence, the Thurston geometrization theorem stated as above was proven by him as a proof of his geometrization conjecture just for the special case of Haken manifolds. \subsection{Thurston's Geometrization Conjectures} William Thurston proposed this more general, geometrization conjecture in 1982 after proving his geometrization theorem. Thurston's geometrization conjecture implies several other conjectures, such as, for example, Thurston's elliptization conjecture, and aslo the Poincar\'e conjecture. This geometrization conjecture can be simply stated as follows: {\bf Thurston Conjecture:} {\em Compact 3-manifolds can be decomposed into submanifolds that have geometric structures.} This geometrization conjecture can be considered as a 3-manifold analogue of the uniformization theorem for 2D-surfaces. William Thurston in 1982, and implies several other conjectures, such as, for example, Thurston's elliptization conjecture, and the Poincar\'e conjecture. Thurston's gemetrization conjecture that was proven later by Grigori Pereleman solved in the affirmative Poincar\'e 's 1904 conjecture. Perelman was awarded in August 2006 the Fields Medal for ``his contributions to geometry and his revolutionary insights into the analytical and geometric structure of the Ricci flow''. (Perelman declined to either accept the Fields Medal award or to appear at the congress where it was supposed to be presented to him.)