Lecture 1

In this course, we focus on compact and orientable 3-manifolds. We ask the

following fundamental questions:

What do all 3-manifolds look like? Can we classify them?

Consider the case of closed orientable surfaces. They are characterized by the

genus. The case g = 0 is the 2-sphere. One can equip it with the round metric. The

Figure 1. Surfaces with genus 0, 1 and 2

case g = 1 corresponds to the two dimensional torus

∼

=

R2/Γ, where Γ is a lattice

subgroup of R2. It admits a naturally induced flat metric from R2. For g ≥ 2 we

can equip the surface Σg of genus g with the hyperbolic metric induced from the

Poincar´ e disk model of H2; that is to say there is a discrete, torsion-free subgroup

Γg of the isometries of

H2

with Σg

∼

=

H2/Γg.

Geometric manifolds

Definition (Homogeneous metric). Let (M, g) be a Riemannian manifold.

The metric g is called homogeneous if the action Isom(M)×M → M is transitive.

Definition (Locally homogeneous metric). A Riemannian metric g on a man-

ifold M is called locally homogeneous if its lifted metric ˜ g on the universal cover

˜

M is homogeneous.

The round metric of the 2-sphere (i.e. g = 0) is homogeneous, but the hyper-

bolic metric of the genus-2 Riemann surface is not. However, the latter is locally

homogeneous.

Definition (Geometric manifolds). A manifold is called geometric if it ad-

mits a finite volume complete locally homogeneous Riemannian metric.

Here is the list of Geometric 3-manifolds by type (see [19]):

(1) Round:

S3

and its finite quotients – lens spaces, dodecahedron spaces –

classification was completed by Hopf.

(2) Flat: T

3

and its finite quotients – these are completely classified.

3

http://dx.doi.org/10.1090/ulect/053/01