# Mathematics in Decision Making, Objectives for Unit Two: Classifying

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