Where can I find a good set of notes discussing main theorems/ideas surrounding non-orientable surfaces?

Question

I'm currently looking at non-orientable surfaces, but know very little about them. Is there are good set of notes that will teach me the classical results surrounding non-orientable surfaces?

Answer 1

There's not a lot to say. Here's a quick summary, for closed compact nonorientable ("N-O") surfaces:

1. No closed N-O surface can be embedded in 3-space; they can all, however, be immersed. "Boy's surface" is a particularly attractive (to me!) immersion of the projective plane into 3-space.

2. Every closed compact N-O surface is a connect-sum of one or more projective planes. The connect sum of two is a Klein bottle, for instance.

3. Every N-O surface contains an embedded Mobius band; a surface with no Mobius bands in it is automatically orientable.

4. The tangent bundle to an N-O surface is nontrivial (on account of item 3); its first Stieffel-Whitney class is nonzero, so the first cohomology group (mod 2) is nontrivial.

5. If the Euler class of a closed compact surface is odd, it must be N-O. If it's even, it may be N-O. For instance, the torus and the Klein bottle both have Euler characteristic 0.

Is there more that you wanted to know? Carlo Sequin of UC Berkeley has made some lovely pictures of various immersions of N-O surfaces; those might be helpful to you.