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I am currently learning about polygonal presentations of surfaces.

In the notation I'm using (following Lee's "Topological Manifolds"), $\langle a, b \ |\ aba^{-1}b^{-1}\rangle$ is a presentation of the torus $\mathbb{T}^2$, and $\langle a,b\ |\ abab \rangle$ is a presentation of the real projective plane $\mathbb{P}^2$. Both of these examples can be thought of as specifying labellings and orientations of the edges of a square, which in turn specify how to glue the edges together to obtain the respective surfaces.

As a fun exercise for myself, I'm trying to list all possible topological spaces (surfaces?) that can result from gluing together the edges of a triangle. I conjecture that the following five four presentations represent all possible such spaces (up to homeomorphism), and also conjecture that they fall into the given homeomorphism classes: $$\langle a \ | \ aaa\rangle \approx \text{?}$$ $$\langle a \ | \ aaa^{-1}\rangle \approx \text{?}$$ $$\langle a, b \ | \ aab\rangle \approx \text{?}$$ $$\langle a, b, c \ | \ abc\rangle \approx \mathbb{D}^2 \text{ (closed disk)}$$

Questions: Are these, in fact, all of them (up to homeomorphism), or are there some that I've missed? Are any two on this list homeomorphic (meaning I've double-counted)? And are there any common descriptions of the homeomorphism classes with question marks? (I realize that "common descriptions" is vague.)

EDIT: By "five" I of course meant "four." That is, $$\langle a, b \ | \ aa^{-1}b\rangle \approx \mathbb{D}^2,$$ which is geometrically clear upon drawing the picture.

Note: These are in fact polygonal presentations, and not group presentations. Because we are dealing with triangles (which are in some sense degenerate), we cannot always read off the fundamental group directly from the polygonal presentation as if it were a group presentation. The example in the "EDIT" illustrates this.

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    With respect to your homeomorphic question, if two spaces are homeomorphic then they have the same fundamental group. So, what are the groups you have here? You have 4 groups - two are isomorphic. Can you work out which two they are? Then work out if the spaces can be homeomorphic...(Also, I'm not sure if "gluing all the edges of a triangle together" makes sense...but I'm no topologist, so it probably does...)2011-07-21
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    Oh, right, the cone and the disk. (Sorry, it's 3 AM here.) Thanks for that. :-)2011-07-21
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    @Jesse Madnick: on a related note, I once had a triangle with these identificactions: let's call A,B,C the vertices starting from the bottom left hand one; then the edges AC and BC towards C were identified and the whole AB edge was identified with C. Maybe this is not something you're interested in because you can't express it in your notation, but maybe it is. :)2011-07-21
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    Your edit isn't what I meant! I was pointing out that if two spaces are homeomorphic then their fundamental groups are isomorphic. You have now edited it so you have $\pi_1(\mathbb{D}^2)=\mathbb{Z}=F_2$, which is a contradiction. Two of your groups are isomorphic to \mathbb{Z}, the other three are pairwise non-isomorphic ($C_3$ (cyclic of order 3), $F_2$ (free on two generators) and the trivial group...).2011-07-21
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    @Swlabr: I don't understand. Are you saying that $\pi_1(\mathbb{D})$ is $\mathbb{Z}$ or $F_2$? Because neither of those sound right to me.2011-07-21
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    You probably got the second one (though it is the trivial group) from the [dunce cap](http://en.wikipedia.org/wiki/Dunce_hat_(topology)). This is homotopy equivalent to the disk, but I suspect that it isn't homeomorphic to the disk.2011-07-21
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    You had said that two things were $\pi_1(\mathbb{D})$, $\mathbb{Z}$ and the free group on two generators. This is clearly a contradiction! As Dylan Moreland has pointed out, the contradiction is more fundamental as it should be trivial...so both groups are wrong!2011-07-21
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    Oh. I looked at Lee's book on Google books; apparently these are "polygonal presentations", and are not explicitly connected to groups. Yikes! I thought you were just being extremely loose with your words.2011-07-21
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    Ohh...I am in the same boat as Dylan...but the presentations given for the torus and the real projective plane *are* the group presentations of their fundamental groups...!2011-07-22
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    @Swlabr: Yes, sorry about that.2011-07-22

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