Explicit Orientation-Reversing Homemorphism of $M_g$

Question

Let $M_g$ be the orientable closed surface of genus $g$. I know that there is an orienation-reversing homeomorphism ($[M] \rightarrow -[M]$, where $[M]$ is fundamental class) $f:M_g \rightarrow M_g$ since $M_g$ is a compact orientable surface, but I am having difficulty finding this explicitly. I think a reflection would do it, but I'm not sure which delta-complex structure to use.

Answer 1

Let $M_g$ be represented as the quotient of a regular $4g$-gon $Q \subset \mathbb{R}^2$ inscribed in the unit circle with one vertex at $(1,0)$, and with pairwise side identifications reading, in clockwise order starting from $(1,0)$, as follows: $$a_1 b_1 a_1^{-1} b_1^{-1} \cdots a_g b_g a_g^{-1} b_g^{-1} $$ The desired homeomorphism $f : M_g \to M_g$ is obtained by restricting to $Q$ the reflection of $\mathbb{R}^2$ across the $x$-axis.