Triangulation of a torus

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drainbramaged
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Triangulation of a torus

Postby drainbramaged » Fri Apr 27, 2012 5:38 am UTC

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

Postby Token » Fri Apr 27, 2012 6:29 am UTC

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

Postby drainbramaged » Fri Apr 27, 2012 10:46 pm UTC

Thanks. That clarification helps a lot with some of the conceptual issues I've been having. You rock!
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magicdex
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Re: Triangulation of a torus

Postby magicdex » Thu Oct 18, 2012 9:06 pm UTC

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

Postby Mindworm » Sat Oct 20, 2012 3:13 pm UTC

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.
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