Geometric 3-manifolds

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Contents

1 Introduction

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
\displaystyle \phi_i:U_i\rightarrow \phi_i\left(U_i\right)\subset X
onto open subsets of 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. 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
\displaystyle 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
\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.

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

Definition 1.1. 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

Theorem 2.1.

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

- Euclidean space: 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=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)

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

- 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\}

- the 3-dimensional solvable Lie group 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).


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.

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