Why can't this triangulate $\mathbb{RP}^2$?

Question

I understand that an actual minimal triangulation of $\mathbb{RP}^2$ has at least 10 2-simplices, but I don't understand why.

Without appealing to the computation of the homology groups of $\mathbb{RP}^2$, why can't the following picture be a triangulation of $\mathbb{RP}^2$? What goes wrong?

Answer 1

Each 2 simplex must be determined uniquely by its vertices. The two 0-1-2 simplexes have the same vertices.

The edges here are OK because they are identified in the projective plane.

• Orientability has nothing to do with it. The thing about an unorientable n-manifold without boundary is it can not be triangulated so that any integer combination of the n simplices forms a cycle. But it can still be triangulated.