Geometric 3-manifolds

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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.There are eight 3-dimensional model geometries:

- 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)}

- 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\}

- 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).

[Thurston1997] Section 3.8

Outline of Proof:

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

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

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

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

Case 2b: \tau is a contact structure. For Y=S^2 one would obtain for G the group of isometries of 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 one obtains G=X=Nil. Namely, G is the subgroup of the group of automorphisms of the standard contact structure dz-xdy=0 on {\mathbb R}^3 consisting of those automorphisms which are lifts of isometries of the x-y-plane. For Y={\mathbb H}^2 one obtains G=X=\widetilde{PSL\left(2,{\mathbb R}\right)}.

Case 3: G_x^\prime=1. Then 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.

3 Invariants


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

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

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

Example 4.2 (Geometrization of mapping tori).

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

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

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

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

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

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

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

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

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

5 Further discussion


6 References

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