Wednesday April 30
Discuss the last week's assignment on surfaces and review.
Last Class: Classification of surfaces almost done!
Review examples: A closed disc, an open disc, a
plane, an annulus- cylinder, a mobius band;
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).
[Activity:Graphs on the torus]
Games and puzzles on the torus and the klein bottle.
Spheres with handles:
Spheres with cross caps
Closed Surfaces: Handles and cross-caps
Visualizations of surfaces by
flattened maps - cut apart models.
A cylinder, a mobius band, the torus, the Klein bottle, the projective
Handles and cross-caps
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 V-E+R = 1
- 2n +1 = 2-2n. For example the torus has euler characteristic 2-2*1=0.
If the surface is orientable - it's Euler characteristic is enough to identify the surface.
If the surface is non-orientable, then is it a sphere with k crosscaps and
n handles, so the euler characteristic is V-E+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
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.