Geometric 3-manifolds
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== Introduction == | == Introduction == | ||
<wikitex>; | <wikitex>; | ||
Let a group $G$ act on a manifold $X$ by homeomorphisms. | 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 $G$. | + | 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. | 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. | ||
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== Construction and examples == | == Construction and examples == | ||
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− | {{beginthm|Theorem}} | + | {{beginthm|Theorem}}There are eight 3-dimensional model geometries: |
- the round sphere: $X=S^3, G=O(4)$ | - the round sphere: $X=S^3, G=O(4)$ | ||
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- the universal covering of the unit tangent bundle of the hyperbolic plane: $G=X=\widetilde{PSL\left(2,{\mathbb R}\right)}$ | - 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 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 $t\rightarrow\left(\begin{matrix}e^t&0\\0&e^{-t}\end{matrix}\right)$.{{endthm}} | ||
+ | {{cite|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 $G=Sol$. | ||
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</wikitex> | </wikitex> | ||
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== Classification/Characterization == | == 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. | ||
+ | |||
<wikitex>; | <wikitex>; | ||
− | ... | + | {{beginthm|Theorem|(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. | ||
+ | {{endthm}} | ||
+ | |||
+ | {{beginthm|Example|(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. | ||
+ | {{endthm}} | ||
+ | |||
</wikitex> | </wikitex> | ||
+ | |||
+ | As the example suggests, the most abundant case ist that of [[Hyperbolic 3-manifolds]]. | ||
== Further discussion == | == Further discussion == | ||
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[[Category:Manifolds]] | [[Category:Manifolds]] | ||
− |
Latest revision as of 14:06, 23 March 2012
This page has not been refereed. The information given here might be incomplete or provisional. |
Contents |
1 Introduction
Let a group act on a manifold by homeomorphisms.
A -manifold is a manifold with a -atlas, that is, a collection of homeomorphisms
onto open subsets of such that all coordinate changes
are restrictions of elements of .
Fix a basepoint and a chart with . Let be the universal covering. These data determine the developing mapIf we change the initial data and , the developing map changes by composition with an element of .
If , analytic continuation along a loop representing gives a chart that is comparable to , since they are both defined at . Let be the element of such that . The mapis a group homomorphism and is called the holonomy of .
If we change the initial data and , the holonomy homomorphisms changes by conjugation with an element of .
A -manifold is complete if the developing map is surjective.
[Thurston1997] Section 3.4
Definition 1.1. A model geometry is a smooth manifold together with a Lie group of diffeomorphisms of , such that:
a) is connected and simply connected;
b) acts transitively on , with compact point stabilizers;
c) is not contained in any larger group of diffeomorphisms of with compact point stabilizers;
d) there exists at least one compact -manifold.
[Thurston1997] Definition 3.8.1
A 3-manifold is said to be a geometric manifold if it is a -manifold for a 3-dimensional model geometry .
2 Construction and examples
Theorem 2.1.There are eight 3-dimensional model geometries:
- the round sphere:
- Euclidean space:
- hyperbolic space:
-
-
- the universal covering of the unit tangent bundle of the hyperbolic plane:
- the Heisenberg group:
- the 3-dimensional solvable Lie group with conjugation .[Thurston1997] Section 3.8
Outline of Proof:
Let be the connected component of the identity of , and let be the stabiliser of . acts transitively and is a closed, connected subgroup of .
Case 1: . Then has constant sectional curvature. The Cartan Theorem implies that (up to rescaling) is isometric to one of .
Case 2: . Let be the -invariant vector field such that, for each , the direction of is the rotation axis of . descends to a vector field on compact -manifolds, therefore the flow of must preserve volume. In our setting this implies that the flow of acts by isometries. Hence the flowlines define a 1-dimensional foliation with embedded leaves. The quotient is a 2-dimensional manifold, which inherits a Riemannian metric such that acts transitively by isometries. Thus has constant curvature and is (up to rescaling) isometric to one of . is a pricipal bundle over with fiber or , The plane field orthogonal to has constant curvature, hence it is either a foliation or a contact structure.
Case 2a: is a foliation. Thus is a flat bundle over . is one of , hence , which implies that .
Case 2b: is a contact structure. For one would obtain for the group of isometries of that preserve the Hopf fibration. This is not a maximal group with compact stabilizers, thus there is no model geometry in this case. For one obtains . Namely, is the subgroup of the group of automorphisms of the standard contact structure on consisting of those automorphisms which are lifts of isometries of the x-y-plane. For one obtains .
Case 3: . Then is a Lie group. The only 3-dimensional unimodular Lie group which is not subsumed by one of the previous geometries is .
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 be a closed, orientable, irreducible, geometrically atoroidal 3-manifold.
a) If is homotopically atoroidal, then it admits an -geometry.
b) If is not homotopically atoroidal, then it admits (at least) one of the seven non--geometries.
Example 4.2 (Geometrization of mapping tori).
Let be an orientation-preserving homeomorphism of the surface of genus .
a) If , then the mapping torus satisfies the following:
1. If is periodic, then admits an geometry.
2. If is reducible, then contains an embedded incompressible torus.
3. If is Anosov, then admits a geometry.
b) If , then the mapping torus satisfies the following:
1. If is periodic, then admits an -geometry.
2. If is reducible, then contains an embedded incompressible torus.
3. If is pseudo-Anosov, then admits an -geometry.
As the example suggests, the most abundant case ist that of Hyperbolic 3-manifolds.
5 Further discussion
...
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