Special triangulations for the torus, Klein bottle and real projective plane

Question

I want to solve the following exercise that asks me to find a apecial type of triangulation for different surfaces but since I am new to triangulations, I am still a bit confused.

Give a triangulation for each the torus, the Klein bottle and the >projective plane, in which the triangles can be colored using 2 colors >such that two adjacent triangles are never coloured equally.

My idea for finding a triangulation of the torus was as follows. I consider the unit square $[0,1] \times [0,1]$. Judging by the picture in this thread
I then should be able to define vertices such that the unit square is decomposed into a collection K of triangles as seen in the picture. But if I start to identify opposite sides I leave the unit spuare and enter the idenitification space. Should I prove that the decomposition defined above does indeed yield a simplicial complex while I'm still in the unit square or after I have done the identifications?

Also assuming I have succeded in proving that K is a simplicial complex, how do I show that the polyhedron |K| is actually homeomorphic to the torus?

Answer 1

You identify the sides first, then draw the triangulation onto the torus. Eg you could draw it onto a donut shape instead of the square. One can normally handwave the step from triangles-on-surface to simplicial complex, you could say something about triangulation forming simplicial set $\to$ isomorphism to simplicial complex $\to$ continuity argument. The answer to the question you linked to explains the conditions for a triangulation drawing to correspond to a simplicial complex.

If you draw the triangulation onto the square you can do something like this:

Ok apologies for the terrible drawing. If you take that triangulation you can make it into a simplicial complex in Euclidean three-space. Here is a sketch of that. Yellow lines correspond to the diagonals above and I’ve only shaded bits of the visible faces. Imagine making a torus from three triangulate prism like shapes, then cutting a diagonal across each side.