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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:

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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)?

Brain
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  • but my second question is different? Why closing it? – Brain Nov 11 '23 at 17:10
  • Regarding your second question, there is no Thm 3.12. The notation $H^*(-)$ means all of cohomology, thought of either as a ring or as a graded group, depending on the context. $H^n(-)$ means the $n$th cohomology group. – John Palmieri Nov 11 '23 at 21:07

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