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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?

Many thanks for your help!

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
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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. –  Matt N. Feb 28 '11 at 16:44
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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

1 Answer 1

up vote 2 down vote accepted

Homology certainly has something to do with orientability, but if you are not yet familiar, I wouldn't recommend diving into the subject for this. You should definitely speak about non-orientable surfaces (e.g. the Klein bottle); this will really help understanding the idea of orientability. Cutting up the Möbius band in front of class is also very insightful.

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thanks for the hint about cutting up the Moebius band in front of the class. I'm not sure though I should mention non-orientable surfaces because the professor explicitly said, "not non-orientable surfaces, orientable ssurfaces". I wonder what to do : ( –  Matt N. Feb 28 '11 at 9:37
    
You could do it in the style of: this is an orientable surface (take a sphere or something), and this is non-orientable (the mobius band). Then explicitly define orientability, and why the sphere satisfies this property, and the mobius band doesn't. –  Thomas Rot Feb 28 '11 at 9:53
    
actually, your first answer pointed me into the right direction. I think the professor expects me to talk about the connection between homology and orientability! I will probably accept your answer later, I just thought I could leave this question open for some more time, to see what other people will suggest. –  Matt N. Feb 28 '11 at 9:58
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@Matt: Great! Do make sure that you pick an easy version for homology (there are many things called homology...). Simplicial homology is the thing you want. A nice reference is the free book "Algebraic Topology" by Allen Hatcher –  Thomas Rot Feb 28 '11 at 13:31
    
Again, many thanks! Hatcher is the book he recommended for the lecture. –  Matt N. Feb 28 '11 at 16:41

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