# Geometric 3-manifolds

## 1 Introduction

Let a group $G$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}G$ act on a manifold $X$$X$ by homeomorphisms.

A $\left(G,X\right)$$\left(G,X\right)$-manifold is a manifold $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_hppiv0$$M$ with a $\left(G,X\right)$$\left(G,X\right)$-atlas, that is, a collection $\left\{\left(U_i,\phi_i\right):i\in I\right\}$$\left\{\left(U_i,\phi_i\right):i\in I\right\}$ of homeomorphisms

$\displaystyle \phi_i:U_i\rightarrow \phi_i\left(U_i\right)\subset X$

onto open subsets of $X$$X$ such that all coordinate changes

$\displaystyle \gamma_{ij}=\phi_i\phi_j^{-1}:\phi_i\left(U_i\cap U_j\right)\rightarrow \phi_j\left(U_i\cap U_j\right)$

are restrictions of elements of $G$$G$.

Fix a basepoint $x_0\in M$$x_0\in M$ and a chart $\left(U_0,\phi_0\right)$$\left(U_0,\phi_0\right)$ with $x_0\in U_0$$x_0\in U_0$. Let $\pi:\widetilde{M}\rightarrow M$$\pi:\widetilde{M}\rightarrow M$ be the universal covering. These data determine the developing map
$\displaystyle D:\widetilde{M}\rightarrow X$
that agrees with the analytic continuation of $\phi_0\pi$$\phi_0\pi$ along each path, in a neighborhood of the path's endpoint.

If we change the initial data $x_0$$x_0$ and $\left(U_0,\phi_0\right)$$\left(U_0,\phi_0\right)$, the developing map $D$$D$ changes by composition with an element of $G$$G$.

If $\sigma\in\pi_1\left(M,x_0\right)$$\sigma\in\pi_1\left(M,x_0\right)$, analytic continuation along a loop representing $\sigma$$\sigma$ gives a chart $\phi_0^\sigma$$\phi_0^\sigma$ that is comparable to $\phi_0$$\phi_0$, since they are both defined at $x_0$$x_0$. Let $g_\sigma$$g_\sigma$ be the element of $G$$G$ such that $\phi_0^\sigma=g_\sigma\phi_0$$\phi_0^\sigma=g_\sigma\phi_0$. The map
$\displaystyle H:\pi_1\left(M,x_0\right)\rightarrow G, H\left(\sigma\right)=g_\sigma$

is a group homomorphism and is called the holonomy of $M$$M$.

If we change the initial data $x_0$$x_0$ and $\left(U_0,\phi_0\right)$$\left(U_0,\phi_0\right)$, the holonomy homomorphisms $H$$H$ changes by conjugation with an element of $G$$G$.

A $\left(G,X\right)$$\left(G,X\right)$-manifold is complete if the developing map $D:\widetilde{M}\rightarrow X$$D:\widetilde{M}\rightarrow X$ is surjective.

[Thurston1997] Section 3.4

Definition 1.1. A model geometry $\left(G,X\right)$$\left(G,X\right)$ is a smooth manifold $X$$X$ together with a Lie group of diffeomorphisms of $X$$X$, such that:

a) $X$$X$ is connected and simply connected;

b) $G$$G$ acts transitively on $X$$X$, with compact point stabilizers;

c) $G$$G$ is not contained in any larger group of diffeomorphisms of $X$$X$ with compact point stabilizers;

d) there exists at least one compact $\left(G,X\right)$$\left(G,X\right)$-manifold.

[Thurston1997] Definition 3.8.1

A 3-manifold is said to be a geometric manifold if it is a $\left(G,X\right)$$\left(G,X\right)$-manifold for a 3-dimensional model geometry $\left(G,X\right)$$\left(G,X\right)$.

## 2 Construction and examples

Theorem 2.1.There are eight 3-dimensional model geometries:

- the round sphere: $X=S^3, G=O(4)$$X=S^3, G=O(4)$

- Euclidean space: $X={\mathbb R}^3, G={\mathbb R}^3\rtimes O(3)$$X={\mathbb R}^3, G={\mathbb R}^3\rtimes O(3)$

- hyperbolic space: $X= H^3, G=PSL\left(2,{\mathbb C}\right)\rtimes {\mathbb Z}/2{\mathbb Z}$$X= H^3, G=PSL\left(2,{\mathbb C}\right)\rtimes {\mathbb Z}/2{\mathbb Z}$

- $X=S^2\times {\mathbb R}, G=O(3)\times \left({\mathbb R}\rtimes {\mathbb Z}/2{\mathbb Z}\right)$$X=S^2\times {\mathbb R}, G=O(3)\times \left({\mathbb R}\rtimes {\mathbb Z}/2{\mathbb Z}\right)$

- $X={\mathbb H}^2\times {\mathbb R}, G=\left(PSL\left(2,{\mathbb R}\right)\rtimes{\mathbb Z}/2{\mathbb Z}\right)\times \left({\mathbb R}\rtimes {\mathbb Z}/2{\mathbb Z}\right)$$X={\mathbb H}^2\times {\mathbb R}, G=\left(PSL\left(2,{\mathbb R}\right)\rtimes{\mathbb Z}/2{\mathbb Z}\right)\times \left({\mathbb R}\rtimes {\mathbb Z}/2{\mathbb Z}\right)$

- the universal covering of the unit tangent bundle of the hyperbolic plane: $G=X=\widetilde{PSL\left(2,{\mathbb R}\right)}$$G=X=\widetilde{PSL\left(2,{\mathbb R}\right)}$

- the Heisenberg group: $G=X=Nil=\left\{\left(\begin{matrix}1&x&z\\0&1&y\\0&0&1\end{matrix}\right):x,y,z\in{\mathbb R}\right\}$$G=X=Nil=\left\{\left(\begin{matrix}1&x&z\\0&1&y\\0&0&1\end{matrix}\right):x,y,z\in{\mathbb R}\right\}$

- the 3-dimensional solvable Lie group $G=X=Sol={\mathbb R}^2\rtimes {\mathbb R}$$G=X=Sol={\mathbb R}^2\rtimes {\mathbb R}$ with conjugation $t\rightarrow\left(\begin{matrix}e^t&0\\0&e^{-t}\end{matrix}\right)$$t\rightarrow\left(\begin{matrix}e^t&0\\0&e^{-t}\end{matrix}\right)$.

[Thurston1997] Section 3.8

Outline of Proof:

Let $G^\prime$$G^\prime$ be the connected component of the identity of $G$$G$, and let $G_x^\prime$$G_x^\prime$ be the stabiliser of $x\in X$$x\in X$. $G^\prime$$G^\prime$ acts transitively and $G_x^\prime$$G_x^\prime$ is a closed, connected subgroup of $SO\left(3\right)$$SO\left(3\right)$.

Case 1: $G_x^\prime=SO\left(3\right)$$G_x^\prime=SO\left(3\right)$. Then $X$$X$ has constant sectional curvature. The Cartan Theorem implies that (up to rescaling) $X$$X$ is isometric to one of $S^3, {\mathbb R}^3, H^3$$S^3, {\mathbb R}^3, H^3$.

Case 2: $G_x^\prime \simeq SO\left(2\right)$$G_x^\prime \simeq SO\left(2\right)$. Let $V$$V$ be the $G^\prime$$G^\prime$-invariant vector field such that, for each $x\in X$$x\in X$, the direction of $V_x$$V_x$ is the rotation axis of $G_x^\prime$$G_x^\prime$. $V$$V$ descends to a vector field on compact $\left(G,X\right)$$\left(G,X\right)$-manifolds, therefore the flow of $V$$V$ must preserve volume. In our setting this implies that the flow of $V$$V$ acts by isometries. Hence the flowlines define a 1-dimensional foliation ${\mathcal{F}}$${\mathcal{F}}$ with embedded leaves. The quotient $X/{\mathcal{F}}$$X/{\mathcal{F}}$ is a 2-dimensional manifold, which inherits a Riemannian metric such that $G^\prime$$G^\prime$ acts transitively by isometries. Thus $Y:=X/{\mathcal{F}}$$Y:=X/{\mathcal{F}}$ has constant curvature and is (up to rescaling) isometric to one of $S^2, {\mathbb R}^2, H^2$$S^2, {\mathbb R}^2, H^2$. $X$$X$ is a pricipal bundle over $Y$$Y$ with fiber ${\mathbb R}$${\mathbb R}$ or $S^1$$S^1$, The plane field $\tau$$\tau$ orthogonal to $\mathcal{F}$$\mathcal{F}$ has constant curvature, hence it is either a foliation or a contact structure.

Case 2a: $\tau$$\tau$ is a foliation. Thus $X$$X$ is a flat bundle over $Y$$Y$. $Y$$Y$ is one of $S^2, {\mathbb R}^2, H^2$$S^2, {\mathbb R}^2, H^2$, hence $\pi_1Y=0$$\pi_1Y=0$, which implies that $X=Y\times {\mathbb R}$$X=Y\times {\mathbb R}$.

Case 2b: $\tau$$\tau$ is a contact structure. For $Y=S^2$$Y=S^2$ one would obtain for $G$$G$ the group of isometries of $S^3$$S^3$ that preserve the Hopf fibration. This is not a maximal group with compact stabilizers, thus there is no model geometry in this case. For $Y={\mathbb R}^2$$Y={\mathbb R}^2$ one obtains $G=X=Nil$$G=X=Nil$. Namely, $G$$G$ is the subgroup of the group of automorphisms of the standard contact structure $dz-xdy=0$$dz-xdy=0$ on ${\mathbb R}^3$${\mathbb R}^3$ consisting of those automorphisms which are lifts of isometries of the x-y-plane. For $Y={\mathbb H}^2$$Y={\mathbb H}^2$ one obtains $G=X=\widetilde{PSL\left(2,{\mathbb R}\right)}$$G=X=\widetilde{PSL\left(2,{\mathbb R}\right)}$.

Case 3: $G_x^\prime=1$$G_x^\prime=1$. Then $X=G^\prime/G_x^\prime=G^\prime$$X=G^\prime/G_x^\prime=G^\prime$ is a Lie group. The only 3-dimensional unimodular Lie group which is not subsumed by one of the previous geometries is $G=Sol$$G=Sol$.

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## 4 Classification/Characterization

A closed 3-manifold is called:

- irreducible, if every embedded 2-sphere bounds an embedded 3-ball,

- geometrically atoroidal, if there is no embedded incompressible torus,

- homotopically atoroidal, if there is no immersed incompressible torus.

Theorem 4.1 (Geometrization).

Let $M$$M$ be a closed, orientable, irreducible, geometrically atoroidal 3-manifold.

a) If $M$$M$ is homotopically atoroidal, then it admits an $H^3$$H^3$-geometry.

b) If $M$$M$ is not homotopically atoroidal, then it admits (at least) one of the seven non-$H^3$$H^3$-geometries.

Example 4.2 (Geometrization of mapping tori).

Let $\Phi:\Sigma_g\rightarrow \Sigma_g$$\Phi:\Sigma_g\rightarrow \Sigma_g$ be an orientation-preserving homeomorphism of the surface of genus $g$$g$.

a) If $g=1$$g=1$, then the mapping torus $M_\Phi$$M_\Phi$ satisfies the following:

1. If $\Phi$$\Phi$ is periodic, then $M_\Phi$$M_\Phi$ admits an ${\mathbb R}^3$${\mathbb R}^3$ geometry.

2. If $\Phi$$\Phi$ is reducible, then $M_\Phi$$M_\Phi$ contains an embedded incompressible torus.

3. If $\Phi$$\Phi$ is Anosov, then $M_\Phi$$M_\Phi$ admits a $Sol$$Sol$ geometry.

b) If $g\ge 2$$g\ge 2$, then the mapping torus $M_\Phi$$M_\Phi$ satisfies the following:

1. If $\Phi$$\Phi$ is periodic, then $M_\Phi$$M_\Phi$ admits an $H^2\times{\mathbb R}$$H^2\times{\mathbb R}$-geometry.

2. If $\Phi$$\Phi$ is reducible, then $M_\Phi$$M_\Phi$ contains an embedded incompressible torus.

3. If $\Phi$$\Phi$ is pseudo-Anosov, then $M_\Phi$$M_\Phi$ admits an $H^3$$H^3$-geometry.

As the example suggests, the most abundant case ist that of Hyperbolic 3-manifolds.

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