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In the following every surface is assumed to be connected. I've read that the commutative monoid of homeomorphism classes of closed surfaces is generated by $P$ (projective plane) and $T$ (torus) subject to the only(!) relation $P^3=PT$. Here the product is given by the connected sum. Now what about the commutative monoid of homeomorphism classes of compact surfaces (with boundary)? Does it also have a nice presentation? I think it is generated by the $P[k]$ and the $T[k]$, where $[k]$ means that $k$ holes have been inserted, and $k$ runs through the natural numbers. What relations do we need? And how do you prove that no others are needed?

Another question, which is rather informal: Do you think that it's worth to read the proofs of these classical classifications? I know the importance of the results, but I suspect that the proofs are just technical.

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2 Answers 2

up vote 8 down vote accepted

I assume you want your surfaces-with-boundary to be compact? Anyway, this cannot be generated by the $P[k]$ and $T[k]$, since you are leaving out the genus 0 surfaces (spheres with holes). Since connect-sum-with-a-disk is the same as removing an open disk, I would work instead with the generators $P$, $T$, and the disk $D$; the $D$ won't interact with the other generators. Since every surface-with-boundary is a surface-minus-some-disks, it seems a presentation is $\langle P,T,D\vert P^3=PT\rangle$.

In answer to your second question: it can be very worthwhile to know the proofs of these classical facts, but that doesn't mean you need to learn the classical proofs. For the classification of surfaces, I have seen Benson Farb give a very nice proof (fitting with the "modern" perspective on mapping class groups etc.) hinging upon the fact that the sphere has the maximal Euler characteristic among surfaces. If I can find any notes of that lecture or a written version, I'll update with a link.

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I'd like to 2nd Church's opinion that proofs of the classification of surfaces can be very informative as the tools you use can be re-used in other circumstances. The key part of the proof is that surfaces can be triangulated (or given a polyhedral decomposition) and are turned into combinatorial objects where you can use homology to finish the classification. Morse theory or transversality (Sard's theorem) is an excellent approach to this if you're dealing with smooth surfaces. These are re-usable tools. –  Ryan Budney Apr 6 '10 at 1:16

I agree with Tom and Ryan that it is worthwhile to learn proofs of the classification of surfaces. I think that the result get a bit of a bad rap since the "standard" combinatorial proof that everyone used to learn (which appears in Seifert-Threlfell's book and Massey's book) is complicated and unenlightening. However, there are now a number of nicer proofs available. Here are a few of my favorites.

1) If you like Morse theory, there is a nice proof in Hirsch's book on differential topology.

2) There is a slick combinatorial proof in Armstrong's book "Basic Topology". I believe that this is the source for the proof mentioned above by Tom Church that Benson Farb likes to give.

3) In Fomenko-Matveev's book "Algorithmic and Computer Methods for Three-Manifolds", there is a nice proof using handle decompositions.

4) There is finally John Conway's "ZIP proof", which was written up by Francis and Weeks in their paper "Conway's ZIP Proof".

All of these proofs assume that the surface has been equipped with either a triangulation (for numbers 2-4) or a smooth structure (for 1). For nice approaches to this, see the answers to my question here. However, when you are first approaching these types of results, I would recommend just assuming that the surfaces can be triangulated or smoothed.

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Whitehead's proof that smooth manifolds admit triangulations is a really beautiful proof that demonstrate very portable techniques. There's a nice write-up in Whitney's "Geometric Integration Theory". –  Ryan Budney Apr 6 '10 at 4:15

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