This should be a simple problem but I started in on it and ran into something I don't understand. Essentially, I want a simplicial decomposition of $D^2\times \mathbb{S}^1$. $D^2$ is the basic 2-simplex (solid triangle), so $D^2\times \mathbb{S}^1$ is a "triangular torus".
Now, see my figure for the attempt at a decomposition (this is naturally pretty hard to draw but I think it's clear enough - the top and bottom should be identified):
The issue for me are the orientations. I start by picking the first simplex to have positive orientation outwards; for consistency that requires the second simplex to have inwards (negative) orientation (first pair of red arrows). Since that simplex has negative orientation, that sets the blue arrows and makes the last simplex have positive orientation again. However, that sets the green arrow, and since the top and bottom should be identified I have an inconsistent orientation (first simplex is supposed to have positive orientation, but in the identification is appears to require negative).
So, either the two faces are identified with reversed orientation (making this thing impossible to draw), or my simplicial decomposition is wrong. I can't seem to find any way to make it correct, so can I get some help? Maybe I need a few more simplices to flip that orientation around?
(This is actually part of a calculation I am trying to verify from Dijkgraaf and Witten, but this is the part I am hung up on).
EDIT: Ok I see one solution is to double the number of simplices, since that would result in a negatively-oriented copy of this; I was distracted by something that was said in the paper to look for 3. But is there a simplier solution?
When you take an oriented 2-simplex $\langle a,b,c\rangle$ you can "multiply" it with a 1-simplex $\langle x,y\rangle$ to get $$ D^2 \times S = \langle a,b,c\rangle \otimes \langle x,y\rangle $$ where $\otimes$ is the complex tensor product according to http://en.wikipedia.org/wiki/Eilenberg%E2%80%93Zilber_theorem which represents your triangle prism as kind of "abstract" CW-complex. For example the simplex-pair $\langle a,b\rangle\otimes\langle x,y\rangle$ represents the rectanlge $ax,bx,by,ay$ as the extrusion of $\langle a,b\rangle$ along edge $\langle x,y\rangle$.
Then you can triangulate this using my algorithm described in http://arxiv.org/abs/1205.5691 However, if you are from the geo-information- or spatial databases-community the algorithm may be "too theoretical" for you. Otherwise you are invited to verify it $-$ I didn't have the time yet to do this but I am quite positive that it works.