Wednesday April 30
Discuss the last week's assignment on surfaces and review.
Last Class: Classification of surfaces almost done!
Euler's Formula for the plane
or the sphere:
Theorem: For any connected network in the plane,
V+R = E + 2.
OR V E+R =2
The number 2 is called the "Euler characteristic
for the plane." ( and the sphere).
Review examples: A closed disc, an open disc, a
plane, an annulus cylinder, a mobius band;
A sphere
A torus
[Activity:Graphs on the torus]
Games and puzzles on the torus and the klein bottle.
Spheres with handles:
Spheres with cross caps
.
Visualizations of surfaces by
flattened maps  cut apart models.
A cylinder, a mobius band, the torus, the Klein bottle, the projective
plane.
Handles and crosscaps
attached to the sphere.
Closed Surfaces: Handles and crosscaps
attached to
the sphere.


A sphere with a handle = a torus

A Sphere with a cross cap = the projective plane

The Topological Classification of "closed surfaces."
Every connected closed and bounded surface is topologically equivalent to a sphere with handles and crosscaps attached.
Proof (Last class!)....
The Euler Characteristic of A Surface.
The classification of surfaces determines the euler characteristic of each surface.
If the surface is orientable, it is a sphere with n handles, so VE+R = 1
 2n +1 = 22n. For example the torus has euler characteristic 22*1=0.
If the surface is orientable  it's Euler characteristic is enough to identify the surface.
If the surface is nonorientable, then is it a sphere with k crosscaps and
n handles, so the euler characteristic is VE+R = 1  (k+2n) +1 =2 2n k.
Notice that a sphere with two cross caps has euler characteristic 0, the
same as the torus. But this was the euler characteristic of the Klein Bottle.
So we should be able to recognize the Klein bottle as a sphere with two cross
caps.
This can be done by a single normalization of one pair of edges with the same orientation.
More topics for today: Continuum Hypothesis?
Smullyan puzzles activities.