Eigenvalues of an induced diffeomorphism on cohomology groups

Question

Let a map $f:S \rightarrow S$ be a diffeomorphism on a closed surface S.

When the map is induced on 2nd (de Rham) cohomology group $$ f^{*}:\text{H}_{DR}^{2}(S) \rightarrow \text{H}_{DR}^{2}(S), $$ where $\text{H}_{DR}^{2}(S)\cong \mathbb{R}$, does the map $f$ have an eigenvalue of 1?