Here is the question I am trying to understand the last part of its solution:
Using the cup product structure, show there is no map $\mathbb R P^n \to \mathbb R P^m$ inducing a nontrivial map $H^1(\mathbb R P^m; \mathbb Z/ 2 \mathbb Z ) \to H^1(\mathbb R P^n; \mathbb Z/ 2 \mathbb Z )$ if $n > m.$ What is the corresponding result for maps $\mathbb C P^n \to \mathbb C P^m.$
Here is the solution to this question I found here:Algebraic Topology Hatcher Chapter 3.2 Problem 3
My question is about this solution is:
Why there the map is a multiplication by a non-zero integer in case of the complex projective space?