Algebraic Topology · Problem 4

Cohomology — Allen Hatcher

Mathematics — Topology · lefschetz-fixed-point-complex-projective

Problem

Apply the Lefschetz fixed point theorem to show that every map

f: \mathbb{C}P^n \to \mathbb{C}P^n

has a fixed point if

n

is even, using the fact that

f^*: H^*(\mathbb{C}P^n; \mathbb{Z}) \to H^*(\mathbb{C}P^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{C}P^n; \mathbb{Z})

. [See Exercise 3 in §2.C for an example of a map without fixed points in this exceptional case.]