I was asked to find out the simplicial homology groups of the torus $T=S^1\times{}S^1$ embedded in $R^3$. I triangulated the torus like this :
Here the $0$-simplices are $\{v_0\}$. $1$-simplices are $\{a,b,c\}$ and the $2$-simplices are $\{D_1,D_2\}$. And I found out the homology groups : $H_0(T)=\mathbb{Z}, H_1(T)=\mathbb{Z}^2,H_2(T)=\mathbb{Z}$
But my teacher said it was wrong, because the triangulation is not correct. According to her it should be:
I don't understand what is wrong with my triangulation of the torus $T$. Can someone please clarify this to me? Thanks! (Excuse the crude pictures!)
Best Answer
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 $a \cup b \cup c$.