= Solution
Take the double cover of $S^2$ branched at $2g+2$ points, with deck <involution> $\iota$. One explicit construction cuts the sphere along $g+1$ disjoint arcs pairing the branch points and cross-glues two copies along the cuts. Its <Euler characteristic> is $2\chi(S^2)-(2g+2)=2-2g$, so it is an oriented closed surface of genus $g$. Locally at a branch point the projection is $z\mapsto z^2$ and the <involution> is $z\mapsto-z$; thus the <involution> is nonidentity and orientation-preserving, also when $g=0$.
The invariant rational <cohomology> of a finite surface action identifies with the <cohomology> of its quotient. For this action, triangulate the quotient with the branch points as vertices and lift the <triangulation>. Quotient <cochains> identify with invariant <cochains>, and averaging over the two deck transformations makes taking invariants exact over $\mathbb Q$. Hence the invariant part of $H^1(S;\mathbb Q)$ is $H^1(S^2;\mathbb Q)=0$. Since $\iota^2=1$, its eigenvalues are $\pm1$ and it is therefore $-I$ on the $2g$-dimensional $H^1$. Its actions on $H^0$ and $H^2$ are both the identity. Thus
$$
\boxed{L(\iota)=1-(-2g)+1=2g+2\ne0.}
$$
<Homotopy> invariance and the result just proved show that \b[every map homotopic to this <involution> has fixed points], for every genus.
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