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Consider the following wrong triangulation of the torus $\mathbb{T}^2$:

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Now for example, the triangle $dbc$ occurs twice and I know that this should not happen. My question is now, why exactly should this not happen? I think it cannot be for the geometric realization of a simplicial complex $K$, that there are two identical triangles mapped to different locations in the space. Is that thinking correct?

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By the definition of triangulation, you need a homeomorphism from a triangle to a geometric triangle, hence if your triangle is mapped into two distinct geometric triangles your map isn't a function and so neither a homeomorphism.