Wednesday April 30

Discuss the last week's assignment on surfaces and review.

Last Class: Classification of surfaces almost done!

1. 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]

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 cross-caps  attached to the sphere.

Closed Surfaces: 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.

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 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.