Triangulation of Torus

algebraic-topology triangulation

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 $a \cup b \cup c$.

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