# The simplest $\Delta$-complex structure on $S^2$

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?

• You can get a simpler structure by modeling the sphere as a CW-complex by using a single 0-cell and a single 2-cell. May 19, 2011 at 18:53
• The OP wants a delta-complex structure, not a cell-complex structure.
– Josh
May 19, 2011 at 20:44
• @Josh: Of course. My comment was not meant to be an answer, but merely meant to illuminate the fact that in other settings things can be simpler. May 19, 2011 at 22:43
• @wchronholm: Ah, that makes sense.
– Josh
May 19, 2011 at 23:55

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).