I'm very new to Topology but we just covered in class an algorithm for calculating the Fundamental Group of a surface given a triangulation. The algorithm goes as follows:
1) Find a spanning tree in the triangulation
2) Add in 2-simplices where two sides are already in the tree
3) Repeat two until you have a maximal simply connected subcomplex
4) Use relations given by the 2-simplices to determine the group.
So given this and the triangulation of S1: S1 Triangulation
I get the spanning tree
Now am I right in thinking that the algorithm doesn't work in the next step because the interior of the circle is not in the surface, so the following is invalid
Whereas the correct continuation would be to pass over step 2 (and 3) to get this
Which would mean that the fundamental group of $S^1$ is just $\langle a\rangle = F^{1} \cong \mathbb{Z}$, and not the trivial group?