I am not quite sure I understand simplicial comlexes/triangulations.
For instance, I think that the below image represents a triangulation for the 2-torus. Am I correct?
Best Answer
The "2-torus" usually means "the thing that looks like the surface of a donut, and is topologically the product $S^1 \times S^1$ of a circle with itself. I think you've got a triangulation of a "2-holed torus".
Certainly it looks as if it's an OK triangulation. The usual problem people get into is trying to find triangulations with very few triangles/vertices, and they end up with two different triangles that have the same three vertices, for instance.
Yours has a fine enough "mesh" that it looks as if that cannot happen, and (assuming you've got the arrows on the edges oriented properly -- I didn't check -- it appears to me to be fine.
As said by @Pece, your computation is correct, because homology can be computed with quite general kinds of complexes.
But it sounds like your teacher wants you to understand the definition of a simplicial complex, and is probably using a definition of a triangulation which requires it to be a simplicial complex.
In a simplicial complex, every simplex is required to be embedded, but none of your 1-simplices $a$, $b$, and $c$ are embedded, because each has its two endpoints attached to the same $0$-simplex $v_0$.
Also, the intersection of any pair of simplices is required to be a simplex, but your 2-simplices $D_1,D_2$ intersect in $a \cup b \cup c$.
The first problem is that the space that one gets by making the identifications in your picture is not the projective plane. In fact it is not a surface, because in the quotient space the image of the four A vertices has no neighborhood homeomorphic to a disc; it does have a neighborhood homeomorphic to two discs identified at their central point, but that is not allowed in a surface.
A simple relabelling of the vertices will correct this issue. Starting from the upper left corner and going clockwise, instead of labelling the vertices as ABCADEABCADE, replace every other A with an X to get ABCXDEABCXDE.
However, even after doing that correction, your picture still does not define a triangulation in the strict sense, and the reason you state is exactly correct: there are two distinct triangles with vertices ABE.
Best Answer
The "2-torus" usually means "the thing that looks like the surface of a donut, and is topologically the product $S^1 \times S^1$ of a circle with itself. I think you've got a triangulation of a "2-holed torus".
Certainly it looks as if it's an OK triangulation. The usual problem people get into is trying to find triangulations with very few triangles/vertices, and they end up with two different triangles that have the same three vertices, for instance.
Yours has a fine enough "mesh" that it looks as if that cannot happen, and (assuming you've got the arrows on the edges oriented properly -- I didn't check -- it appears to me to be fine.