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I am a third year maths student. I am self-studying a course on surfaces. I have some questions and would really appreciate it if people can help me.

  1. What exactly is a connected sum? According to my lecture notes, for two closed (compact) surfaces, if we remove a closed disc from each then take a homeomorphism from the boundary of one disc to the other, we get another surface. My question is, what do we mean by 'closed disc' on a surface? Do we mean some kind of ball? But what metric are we using then?

  2. When forming connected sum, does it matter if we don't remove a disc but something homeomorphic to a disc? Like a square or triangle for example. I don't think it matters.

  3. It is stated that removing a closed disc from the projective space gives the Möbius band. I don't see why this is the case. Can we do it by the edge identification diagrams?

  4. It is stated that the connected sum of two projective spaces is 'obviously' the Klein bottle. I think this is very non-obvious. How can I see this with edge identification diagrams?

Any help would be appreciated, I am learning from these notes if anyone is interested:

http://www.maths.ox.ac.uk/system/files/coursematerial/2012/2377/29/hitchin1.pdf

Lastly, I am likely to have many more questions about the geometry of surfaces. If anyone is interested in tutoring/helping me, please say so, I am looking for a teacher. I will pay you.

azimut
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John Michael
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  • for three and four, i recommend you look at my comments, mainly the imgur album, on this question http://math.stackexchange.com/questions/418048/the-set-of-lines-in-mathbbr2-is-a-mobius-band/427414#427414 – citedcorpse Jul 05 '13 at 19:32

1 Answers1

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  1. Surface already means that for every point you have charts (homeomorphisms, or maybe diffeomorphisms to some $\mathbb{R}^n$). This already tells you how to draw balls because you can draw them on the $\mathbb{R}^n$ first and then map them to your surface.

  2. See 1.

  3. and 4. Yes you can see it from the edge identification diagrams. Remember that you don't need to get exactly the same diagrams, but that you can get from one to the other by cutting new edges and glueing sides that can be identified. I think is a nice not-so-hard puzzle that you should try doing. It is like playing with a Tangram.

You can find the pictures here. but do try it before looking it up.

OR.
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  • For 2., what if there are more than one homeomorphisms from the surface to R^2? The notes I used cut out a non-circular shape from the edge identification diagram (a square). Is this equivalent to cutting out a disc? – John Michael Jul 06 '13 at 04:48
  • Square and disc are homeomorphic. You care about figures up to homeomorphism. – OR. Jul 06 '13 at 04:53