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I think my reasoning is correct, but I want to run through it here because having the right intuition will make similar problems easier in future.

A 2-simplex is homeomorphic to a closed disc, and a closed disc is homeomorphic to a hemisphere, so we can "build" $S^2$ out of two 2-simplices. However we need to have 3 specified vertices, say $v_0, v_1$ and $v_2$, on the circumference where the two hemispheres meet. This also means three 1-simplices joining these vertices.

Is this the simplest $\Delta$-structure possible?

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    @wchronholm: Ah, that makes sense.2011-05-19

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You need at least two $2$-simplices, but you can glue them up in another way to get an equally simple structure. Take a simplex and glue two of its sides together to get a cone. Do this for another simplex, and then glue the boundary circles together. This is a $\Delta$ complex structure on the sphere with 3 vertices 3 edges and 2 faces, just like yours, but glued together differently.

To see that you can't get away with just one $2$-simplex, you just have to notice that there's no way of gluing the sides of a triangle together to get a sphere. (or even a surface).

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    +1 Great, this is exactly what I wanted to know. That there isn't anything simpler than 3 vertices, 3 edges and 2 faces was the hunch I had. Interesting alternative structure there too..2011-05-19