Originally Posted 12-24-2011
Let’s continue to think about the structure of this course, and move on to the second unit, which is about classifying compact surfaces.
Learning Objectives for the Classification of Compact Surfaces
This section of the course is going to be very hands-on. The technical details of a proper proof are very challenging–especially the fact that a compact, Hausdorff, second-countable 2-manifold has a triangulation (Rado’s theorem). I will gloss over this bit of unpleasantness. I haven’t yet read all of the different approaches to this theorem, but I am leaning towards Conway’s ZIP proof.
Here are the list of things that students should master:
- The idea of a surface, with examples
- one-sidedness vs two-sidedness
- the concept of homeomorphism as distinct from ambient isotopy. I’ll likely not use those words, and instead try to use these:
- cut and paste (or sewing pattern) equivalence
- deformation equivalence
- sewing operations:
- connected sum
- punching holes
- adding handles
- adding cross handles
- adding cross caps
- admissable graphs on a surface and the Euler Characteristic
- The classification
I’ll be happy to get through all of this in five or six weeks. Spring Break will occur somewhere in this unit, so I’ll have to look for a good way to break things up.
It is important to me that we spend a lot of time learning how to draw the pictures and visualize what is going on. We might spend some time doing cut and paste activities in class, and we will definitely spend some time drawing. I will rely a bit on George Francis’s work for the drawings.
I sincerely hope that this unit will “draw them in” to the beauty of topology. [fozzie bear]Ha! Ha! I am so funny![/fozzy bear]