Geometric 3-manifolds
Contents |
1 Introduction
Let a group act on a manifold
by homeomorphisms.
![\left(G,X\right)](/images/math/d/1/2/d12d43ff00260ab4aa5b4665d805372d.png)
![M](/images/math/8/9/f/89f5942bc37eb2131578d77a79571c59.png)
![\left(G,X\right)](/images/math/d/1/2/d12d43ff00260ab4aa5b4665d805372d.png)
![\left\{\left(U_i,\phi_i\right):i\in I\right\}](/images/math/d/3/5/d358c367560471419ceef9090ab39c5a.png)
![\displaystyle \phi_i:U_i\rightarrow \phi_i\left(U_i\right)\subset X](/images/math/d/5/6/d567b7fd32b2e1bb323cf0bb07622fbd.png)
![X](/images/math/4/7/4/474e6c59d39ab2b9f9eb79ab75b9da90.png)
![\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)](/images/math/f/d/d/fdd2cbbffbadcba3b74769ea497da37c.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![x_0\in M](/images/math/9/e/7/9e7bf952aaba902d415d16423c422448.png)
![\left(U_0,\phi_0\right)](/images/math/6/5/3/65370a9bbfc66548177a1c63cec4e6c3.png)
![x_0\in U_0](/images/math/a/1/a/a1ae33cce6e4bfbcfded9c6729ca1d3f.png)
![\pi:\widetilde{M}\rightarrow M](/images/math/3/3/5/335502576d733a4d492983b526648820.png)
![\displaystyle D:\widetilde{M}\rightarrow X](/images/math/b/b/d/bbd89d125039aa7d6ba8da99a78c6085.png)
![\phi_0\pi](/images/math/e/c/7/ec78054d726a6f51e03848dba68d3002.png)
If we change the initial data and
, the developing map
changes by composition with an element of
.
![\sigma\in\pi_1\left(M,x_0\right)](/images/math/3/5/9/3598848098baf4cd0ffbeb726a6dbfc7.png)
![\sigma](/images/math/b/a/e/bae0e2c6b9e87dee9762427e8eb6ffe0.png)
![\phi_0^\sigma](/images/math/5/c/8/5c87c318951c72e1c766f55b1c1bfc1e.png)
![\phi_0](/images/math/0/0/2/002021b06bca528462037b69dac83bd1.png)
![x_0](/images/math/b/0/d/b0dc15d502d67805b23c816770a166d1.png)
![g_\sigma](/images/math/a/b/c/abcbbed52eecd20fabcc4e88ea1e46e4.png)
![G](/images/math/e/3/3/e338776be8d95f5e8ff908e86c07e630.png)
![\phi_0^\sigma=g_\sigma\phi_0](/images/math/6/4/1/64145cd5e6fba99dcb0bb3a6a0921949.png)
![\displaystyle H:\pi_1\left(M,x_0\right)\rightarrow G, H\left(\sigma\right)=g_\sigma](/images/math/c/c/9/cc9eb0f0d9cce674d77e6a960dba19c4.png)
is 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:
![G=X=Sol={\mathbb R}^2\rtimes {\mathbb R}](/images/math/c/6/f/c6ff686bfd239ee32df0f2f2082a5e9d.png)
![t\rightarrow\left(\begin{matrix}e^t&0\\0&e^{-t}\end{matrix}\right)](/images/math/0/3/4/03434dae95d58e8376c9e3150d1c8d3b.png)
[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
...
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.
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
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