Here is the question I am trying to understand 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 an answer I found online:
Here are my questions about it:
1- In the third line, why the induced map $f^*$ has to be the identity map?
2- Does Allen Hatcher book "Algebraic Topology" has the same notion for cohomology groups and cohomology rings (Thm. 3.12)?