Geometric 3-manifolds
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 homeomorphismsIf 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 thise 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
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4 Classification/Characterization
A closed 3-manifold is called:
- irreducibel, if every embedded 2-sphere bounds a 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 a -geometry.
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
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