Apply the Lefschetz fixed point theorem to show that every map \(f: \mathbb{CP}^n \to \mathbb{CP}^n\) has a fixed point if \(n\) is even, using the fact that \(f^*: H^*(\mathbb{CP}^n; \mathbb{Z}) \to H^*(\mathbb{CP}^n; \mathbb{Z})\) is a ring homomorphism. When \(n\) is odd show there is a fixed point unless \(f^*(\alpha) = -\alpha\), for \(\alpha\) a generator of \(H^2(\mathbb{CP}^n; \mathbb{Z})\).
Solution
Let \(\alpha\in H^2(\mathbb{CP}^n;\mathbb Z)\) be the standard generator. Since \(f^*\) is a ring map, there is an integer \(d\) such that \[ f^*(\alpha)=d\alpha, \qquad f^*(\alpha^j)=d^j\alpha^j. \] Each even-dimensional cohomology group is infinite cyclic and the odd groups vanish. Hence the Lefschetz number is \[ L(f)=1+d+d^2+\cdots+d^n. \] If \(n\) is even, this integer is never zero. Indeed for \(d\ge0\) it is positive; for \(d<0\), writing \(d=-r\) with \(r\ge1\), one gets \[ 1-r+r^2-\cdots+r^n=\frac{r^{n+1}+1}{r+1}>0. \] Thus \(L(f)\ne0\), so the Lefschetz fixed point theorem gives a fixed point.
If \(n\) is odd, then \[ 1+d+\cdots+d^n=0 \] can occur only for \(d=-1\): for \(d\ge0\) the sum is positive, and for \(d=-r<0\), \[ 1-r+r^2-\cdots-r^n=(1-r)(1+r^2+\cdots+r^{n-1}), \] which vanishes exactly when \(r=1\). Therefore every map has a fixed point unless \[ f^*(\alpha)=-\alpha. \]