State the Lefschetz fixed point theorem.

Let $X$ be an orientable surface of genus $g$ (which you may suppose has a triangulation), and let $f: X \rightarrow X$ be a continuous map such that

1. $f^{3}=\operatorname{Id}_{X}$,

2. $f$ has no fixed points.

By considering the eigenvalues of the linear map $f_{*}: H_{1}(X ; \mathbb{Q}) \rightarrow H_{1}(X ; \mathbb{Q})$, and their multiplicities, show that $g$ must be congruent to 1 modulo 3 .