Past exam of the mathematics course of the University of Cambridge / 2005 / iii / Paper 13 / 4 / i

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 $\pm 1$ and it is therefore $-I$ on the $2g$-dimensional $H^1$. Its actions on $H^0$ and $H^2$ are both the identity. ThusHomotopy invariance and the result just proved show that every map homotopic to this involution has fixed points, for every genus.