Geometric 3-manifolds

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1 Introduction

Let a group $ == Introduction == <wikitex>; Let a group $G$ act on a manifold $X$ by homeomorphisms. A $\left(G,X\right)$-manifold is a manifold $M$ with a $\left(G,X\right)$-atlas, that is, a collection $\left\{\left(U_i,\phi_i\right):i\in I\right\}$ of homeomorphisms $$\phi_i:U_i\rightarrow \phi_i\left(U_i\right)\subset X$$ onto open subsets of $X$ such that all coordinate changes $$\gamma_{ij}=\phi_i\phi_j^{-1}:\phi_i\left(U_i\cap U_j\right)\rightarrow \phi_j\left(U_i\cap U_j\right)$$ are restrictions of elements of $G$. Fix a basepoint $x_0\in M$ and a chart $\left(U_0,\phi_0\right)$ with $x_0\in U_0$. Let $\pi:\widetilde{M}\rightarrow M$ be the universal covering. These data determine the developing map $$D:\widetilde{M}\rightarrow X$$ that agrees with the analytic continuation of $\phi_0\pi$ along each path, in a neighborhood of the path's endpoint. If we change the initial data $x_0$ and $\left(U_0,\phi_0\right)$, the developing map $D$ changes by composition with an element of $G$. If $\sigma\in\pi_1\left(M,x_0\right)$, analytic continuation along a loop representing $\sigma$ gives a chart $\phi_0^\sigma$ that is comparable to $\phi_0, since they are both defined at $x_0$. Let $g_\sigma$ be the element of $G$ such that $\phi_0^\sigma=g_\sigma\phi_0$. The map $$H:\pi_1\left(M,x_0\right)\rightarrow G, H\left(\sigma\right)=g_\sigma$$ is a group homomorphism and is called the holonomy of $M$. If we change the initial data $x_0$ and $\left(U_0,\phi_0\right)$, the holonomy homomorphisms $H$ changes by conjugation with an element of $G$. A $\left(G,X\right)$-manifold is complete if the developing map $D:\widetilde{M}\rightarrow X$ is surjective. </wikitex> == Construction and examples == <wikitex>; ... </wikitex> == Invariants == <wikitex>; ... </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}}  [[Category:Manifolds]] {{Stub}}G$ act on a manifold $X$ by homeomorphisms.

A $\left(G,X\right)$ $\left(G,X\right)$ -manifold is a manifold

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$M$ 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

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

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

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

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

$\displaystyle D:\widetilde{M}\rightarrow X$ $\displaystyle D:\widetilde{M}\rightarrow X$

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

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

$\displaystyle H:\pi_1\left(M,x_0\right)\rightarrow G, H\left(\sigma\right)=g_\sigma$ $\displaystyle H:\pi_1\left(M,x_0\right)\rightarrow G, H\left(\sigma\right)=g_\sigma$

is a group homomorphism and is called the holonomy of

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$M$ .

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

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

[Thurston1997] Section 3.4

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

a) $X$ is connected and simply connected;

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

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

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

[Thurston1997] Definition 3.8.1

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

2 Construction and examples

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3 Invariants

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4 Classification/Characterization

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5 Further discussion

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6 References

[Thurston1997] W. P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton University Press, Princeton, NJ, 1997. MR1435975 (97m:57016) Zbl 0873.57001

This page has not been refereed. The information given here might be incomplete or provisional.

@@ Line 21: / Line 21: @@
 If we change the initial data $x_0$ and $\left(U_0,\phi_0\right)$, the developing map $D$ changes by composition with an element of $G$.
-If $\sigma\in\pi_1\left(M,x_0\right)$, analytic continuation along a loop representing $\sigma$ gives a chart $\phi_0^\sigma$ that is comparable to $\phi_0, since they are both defined at $x_0$. Let $g_\sigma$ be the element of $G$ such that $\phi_0^\sigma=g_\sigma\phi_0$. The map $$H:\pi_1\left(M,x_0\right)\rightarrow G, H\left(\sigma\right)=g_\sigma$$
+If $\sigma\in\pi_1\left(M,x_0\right)$, analytic continuation along a loop representing $\sigma$ gives a chart $\phi_0^\sigma$ that is comparable to $\phi_0$, since they are both defined at $x_0$. Let $g_\sigma$ be the element of $G$ such that $\phi_0^\sigma=g_\sigma\phi_0$. The map $$H:\pi_1\left(M,x_0\right)\rightarrow G, H\left(\sigma\right)=g_\sigma$$
 is a group homomorphism and is called the holonomy of $M$.
@@ Line 27: / Line 27: @@
 A $\left(G,X\right)$-manifold is complete if the developing map $D:\widetilde{M}\rightarrow X$ is surjective.
+{{cite|Thurston1997}} Section 3.4
+{{beginthm|Definition}}
+A model geometry $\left(G,X\right)$ is a smooth manifold $X$ together with a Lie group of diffeomorphisms of $X$, such that:
+a) $X$ is connected and simply connected;
+b) $G$ acts transitively on $X$, with compact point stabilizers;
+c) $G$ is not contained in any larger group of diffeomorphisms of $X$ with compact point stabilizers;
+d) there exists at least one compact $\left(G,X\right)$-manifold.
+ {{endthm}}
+{{cite|Thurston1997}} Definition 3.8.1
+A 3-manifold is said to be a geometric manifold if it is a $\left(G,X\right)$-manifold for a 3-dimensional model geometry $\left(G,X\right)$.
 </wikitex>

Geometric 3-manifolds

Revision as of 10:35, 8 June 2010

Contents

1 Introduction

2 Construction and examples

3 Invariants

4 Classification/Characterization

5 Further discussion

6 References

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