Triangulation of a torus

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drainbramaged
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Triangulation of a torus

Hi all! Why is the subdivision below not a proper triangulation for a torus? This has got me stumped.
Thanks!

Spoiler:
Is it because the two inner triangles who share a right-angle vertex in the middle share all three points?
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Token
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Re: Triangulation of a torus

Exactly right. Check out the definition of a triangulation - it's a homeomorphism between a simplicial complex and whatever space you're triangulating. A simplicial complex is just a glued-together collection of simplices, which are just triangles / tetrahedrons / higher-dimesional versions of the same. Now, on the surface of a torus, two triangles can share all three vertices (as demonstrated above). But simplices live in Rn, where such a thing is impossible. Therefore, what you've mapped into that torus isn't really a simplicial complex, so it's not really a triangulation.
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drainbramaged
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Re: Triangulation of a torus

Thanks. That clarification helps a lot with some of the conceptual issues I've been having. You rock!
For those who understand, no explanation is necessary. For those who do not understand, no explanation is possible.

magicdex
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Joined: Thu Oct 18, 2012 9:01 pm UTC

Re: Triangulation of a torus

http://i1207.photobucket.com/albums/bb4 ... sphere.png

On a similar theme, can anyone tell me why this is not a triangulation of a 2-sphere?

Mindworm
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Re: Triangulation of a torus

That one seems valid. Of course you do need the three lines and three points seperately, but I think they were implied in the picture.
edit: Weird. In researching why there was even a question in this thread, I think I stumbled upon two non-equivalent definitions of simplicial (cell) complexes. The one I was using (and considered the only one) is essentially simplices glued together at their edges. The one most people seem to be using involves doing this in R^n for some n and preserving the "flatness" of the simplices. In that case, the second one suffers from the same problem as the first one, the triangles coincide.
In my defense, the definition I am using produces easier triangulations (two triangles give you the sphere, the torus, rp2, the klein bottle) while still easily calculating the homology and cohomology by interpreting it as a chain complex.
The cake is a pie.