I'm giving a presentation about orientable surfaces (as a student project) and I was wondering what I should talk about. The presentation should be 20-30 min long.

I've been thinking maybe something like this:

1) definition of a surface

2) what does orientable mean (orientable vs. oriented)

3)

And then I'm a bit lost. In the general topology course last year, classification of compact surfaces and covering maps were mentioned but I don't much about either and I don't know if it has anything to do with orientability of surfaces.

In this class, algebraic topology, we're going to learn about cell complexes and homology, but I don't think that has anything to do with orientability either.

Question: what is an interesting fact or theorem that I should definitely mention in this presentation?

Edit:

I did

1) definition of surface and manifold

2) definition of orientable (with paper moebius strip)

3) classification of compact orientable surfaces

Just in case anyone reads this post later, they might use this to get inspiration. I had also prepared material about homology but the professor said homology was going to be discussed towards the very end of the lecture so I did not present what I had prepared.

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I think the classification of compact connected orientable surfaces is an appropriate topic. – Grumpy Parsnip Feb 28 '11 at 11:37
It also wouldn't hurt to ask your professor directly what he or she is expecting! – Grumpy Parsnip Feb 28 '11 at 15:38
@Jim: Yes, I was going to do that but I don't like people who ask questions without first thinking so I wanted to first think about what he could be expecting and then ask him if what I'm planning to do is meeting his expectations. Thanks for the hint about the classification of compact orientable surfaces! I think I'll post what I'll be doing when I finish the preparation. – Rudy the Reindeer Feb 28 '11 at 16:44
It seems silly to talk about the notion of orientability without mentioning that non-orientable surfaces exist. The Moebius band would be an efficient example. Another topic you could mention is the Alexander-Schoenflies theorem, that a there are no "exotic" ways of putting a sphere in 3-dimensional space. In contrast, there are infinitely-many "knotted" ways of putting a genus $n$ orientable surface in space (for each $n$) provided $n \geq 1$. – Ryan Budney Mar 4 '11 at 15:58
A nice definition of orientable surface would be that it does not contain a Moebius band. Similarly, for an orientable surface you could say a neighbourhood of a simple closed path has to "look like" an annulus. – Ryan Budney Mar 4 '11 at 15:59