Geometric 3manifolds
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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 3manifold is said to be a geometric manifold if it is a manifold for a 3dimensional model geometry .
2 Construction and examples
Theorem 2.1.There are eight 3dimensional 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 3dimensional 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 1dimensional foliation with embedded leaves. The quotient is a 2dimensional 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 xyplane. For one obtains .
Case 3: . Then is a Lie group. The only 3dimensional 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 3manifold is called:
 irreducible, if every embedded 2sphere bounds an embedded 3ball,
 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 3manifold.
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 nongeometries.
Example 4.2 (Geometrization of mapping tori).
Let be an orientationpreserving 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 pseudoAnosov, then admits an geometry.
5 Further discussion
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6 References
 [Thurston1997] W. P. Thurston, Threedimensional geometry and topology. Vol. 1, Princeton University Press, Princeton, NJ, 1997. MR1435975 (97m:57016) Zbl 0873.57001