Geometric 3-manifolds

1 Introduction

Let a group $G$$<!-- COMMENT: To achieve a unified layout, along with using the template below, please OBSERVE the following: besides, $...$ and $$...$$, you should use two environments: - For statements like Theorem, Lemma, Definition etc., use e.g. {{beginthm|Theorem 1|(Milnor)}} ... ... ... {{endthm}}. - For references, use e.g. {{cite|Milnor1958b}}. END OF COMMENT --> == Introduction == <wikitex>; Let a group $G$ act on a manifold $X$ by homeomorphisms. A $\left(G,X\right)$-manifold is a manifold $M$ with a $\left(G,X\right)$-atlas, that is, a collection $\left\{\left(U_i,\phi_i\right):i\in I\right\}$ of homeomorphisms $$\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$$ is a group homomorphism and is called the holonomy of $M$. If we change the initial data $x_0$ and $\left(U_0,\phi_0\right)$, the holonomy homomorphisms $H$ changes by conjugation with an element of $G$. A $\left(G,X\right)$-manifold is complete if the developing map $D:\widetilde{M}\rightarrow X$ is surjective. </wikitex> == Construction and examples == <wikitex>; ... </wikitex> == Invariants == <wikitex>; ... </wikitex> == Classification/Characterization == <wikitex>; ... </wikitex> == Further discussion == <wikitex>; ... </wikitex> == References == {{#RefList:}} <!-- Please modify these headings or choose other headings according to your needs. --> [[Category:Manifolds]] {{Stub}}G$ act on a manifold $X$$X$ by homeomorphisms.

A $\left(G,X\right)$$\left(G,X\right)$-manifold is a manifold $M$$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$$x_0$ and $\left(U_0,\phi_0\right)$$\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$$\phi_0^\sigma$ that is comparable to $\phi_0, since they are both defined at$$\phi_0, since they are both defined at$x_0$. Let$$. Let$g_\sigma$be the element of$$be the element of$G$such that$$such that$\phi_0^\sigma=g_\sigma\phi_0$. 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 $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.

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

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