Extend simplicial homeomorphism in a PL surface

Question

Let $S$ be a connected PL closed surface. How can I show that, given a 2-simplex $\Delta$ in $S$ and a simplicial homeomorphism $g:\Delta\to \Delta$ that preserves orientation, this can be extended to a simplicial homeomorphism $h:S\to S$ ?

EDIT:

Answer 1

Since it preserves orientation, your homeomorphism is going to be a rotation of the simplex. Take a narrow annulus $A$ surrounding $\Delta$ with one boundary component $\partial_0A$ equal to the boundary of the simplex $\Delta$. Now we want to extend $g$ from $\partial_0A$ to $A$ in such a way that it restricts to the identity on $\partial _1A$. To do this note that $g|_{\partial_0A}$ is a rotation, and hence isotopic to the identity. Use the annulus $A$ to actually parametrize this isotopy. Now extend to all of $S$.