TITLE: Every continuous map $f:\mathbb{C}P(2) \to \mathbb{C}P(2)$ has a fixed point, without Lefschetz theorem.
QUESTION [5 upvotes]: I would like to know if there is a nice proof of the fact that every continuous map $f:\mathbb{C}P(2) \to \mathbb{C}P(2)$ has a fixed point, without use of the Lefschetz fixed point theorem.

REPLY [5 votes]: I don't understand the first paragraph of Connor's answer at all, but assuming it checks out, in general the total Chern class of $\mathbb{CP}^n$ is
$$(1 + \alpha)^{n+1} = \sum_{k=0}^n {n+1 \choose k} \alpha^k \in \mathbb{Z}[\alpha]/\alpha^{n+1}$$
(note that the $k = n+1$ term vanishes). If the (complex) tangent bundle of $\mathbb{CP}^n$ has a (complex) line subbundle then the total Chern class must factor as
$$(1 + c \alpha)(1 + c_1 \alpha + \dots + c_{n-1} \alpha^{n-1})$$
and since we never get a coefficient of $\alpha^{n+1}$ or higher, the problem of determining whether this is possible is equivalent to the problem of determining when $(1 + \alpha)^{n+1} - \alpha^{n+1}$ has a linear factor of the form $(1 + c \alpha)$, as an ordinary polynomial.
It will be easier to reverse the order of the coefficients: this is equivalent to determining when $\frac{(x + 1)^{n+1} - 1}{x}$ has a linear factor of the form $(x + c)$, where $c$ is an integer. This implies
$$(1-c)^{n+1} = 1$$
and if $n$ is even this gives $c = 0$, but $x + c = x$ is not a factor of the above polynomial because its constant coefficient is $n+1$ (this coefficient corresponds to the top Chern class and hence to the Euler characteristic $\chi(\mathbb{CP}^n) = n+1$ so we really are using that the Euler characteristic doesn't vanish). We conclude:

Claim: $\mathbb{CP}^{2m}$ has the fixed point property.

This is usually proven by Lefschetz (as far as I know anyway). If $n$ is odd then this gives either $c = 0$ or $c = 2$ so we don't get a contradiction, and I guess the existence of fixed-point-free maps in this case implies that the tangent bundle has a line subbundle with first Chern class $2$. I wonder if anyone knows a more explicit description of it.