Self-Homeomorphism of an orientable surface with boundary

Question

How can I show that a self-homeomorphism of an orientable surface with boundary that fixes identically a boundary component is orientation-preserving?

Answer 1

If you really want to work in the topological category then the answer of @JasonDeVito is the only one available.

If you are willing to work in the differentiable category, let me denote the surface and its diffeomorphism by $f : S \to S$.

Pick $p \in \partial S$ and pick a nonzero $v \in T_p \partial S$. By your assumption we have $D_pf(v)=v$.

The $1$-dimensional subspace $T_p \partial S \subset T_p S$ separates the 2-dimensional vector space $T_p S$ into two half spaces, one denoted $H_{in}$ consisting of vectors that point inward, and the other $H_{out}$ consisting of vector that point outward. Since $f : S \to S$ is a diffeomorphism it follows that $D_p f(H_{in}) = H_{in}$.

The two equations $D_p f(v)=v$ and $D_p f (H_{in}) = H_{in}$ imply that $D_p f$ preserves the orientation of $T_p S$. Therefore $f$ preserves orientation.