Problem from Rotman's Algebraic Topology book suppose $ n > m $ and $ i : RP^m \to RP^n $ is the natural imbedding.Then show that $ i^* : H^q(RP^n ; Z_2) \to H^q(RP^m ; Z_2)$ is an isomorphism for all $ q < m+1$
this is a problem from Rotman's Algebraic Topology book. i think if i can show that this is true for $ q = 1$ ,then we are done. but i can't show that.
 A: Let $p$ be a point in $\mathbb{R}P^n\setminus i(\mathbb{R}P^{n-1})$ and let $B_p$ be a small ball around $p$ which does not intersect $i(\mathbb{R}P^{n-1})$. Consider the Mayer Vietoris sequence for the pair of subspaces $A=\mathbb{R}P^n\setminus\{p\}$ and $B=B_p$. This sequences looks like
$$\cdots\to H^{q-1}(A\cap B) \to H^q(\mathbb{R}P^n)\stackrel{f^*}{\to} H^q(A)\oplus H^q(B)\stackrel{p^*}{\to} H^q(A\cap B)\to H^{q+1}(\mathbb{R}P^n)\to \cdots$$
Show and then use the following facts to reach the conclusion in the question for $m=n-1$, and then use an inductive argument to extend this to all $m<n$


*

*$B$ is contractible.

*$A$ deformation retracts onto $i(\mathbb{R}P^{n-1})$ (what does this say about the relationship between $i^*$ and $f^*$?).

*$A\cap B$ is homotopy equivalent to $S^{n-1}$.

*The map $p^*$ is really the same as the map $\times2^*\colon H^q(\mathbb{R}P^{n-1}) \to H^q(S^{n-1}) $ induced by the double cover $\times2\colon S^{n-1}\to \mathbb{R}P^{n-1}$ (this comes from the cellular structure of $\mathbb{R}P^n$ with one cell in each dimensions and the relevant attaching maps).

A: Have you tried using cellular homology (the inclusion is a cellular map if you take the usual CW-structures)?
