For $n\geq 2$, any continuous map $f:\mathbb{C}P^n\rightarrow S^2$ induces the zero map on $H_2(*)$ I am working through an old qualifying exam from another university.  My course did not cover as much material as what is on this test (e.g. we did not cover cohomology).  So I am just working through the problems which look like fair game for the material covered in my course. I tried the following problem (because it looks like I should now how to do it), and I am very stuck on it:
For $n\geq 2$, any continuous map $f:\mathbb{C}P^n\rightarrow S^2$ induces the zero map on $H_2(*)$.
I know that $H_2(\mathbb{C}P^n)=\mathbb{Z}$ and $H_2(S^2)=\mathbb{Z}$ as well.  So it is not obvious yet that $f_*$ is the zero map.  I tried looking at the long exact sequence for the pair $(\mathbb{C}P^n, S^2)$ and computed the relative homology groups; all of the seemingly relevant ones are trivial (i.e. $i=1,2,3$).  So the long exact sequence doesn't seem to give me any more information.  I am not sure what I am missing here.   
 A: The cohomology ring of $\Bbb{CP}^n$ is $\Bbb Z[x]/(x^{n+1})$, with $|x|=2$. Any map $f: \Bbb{CP}^n \to S^2$ induces zero on cohomology when $n>1$, because if $z \in H^2(S^2)$ is a generator, then $0 = f^*(z^2) = f^*(z)^2$, so $f^*(z)$ must be zero.
Now use the fact that the universal coefficient theorem is natural to see that the induced map on $H_2$ is also zero. (I don't really want to draw the diagram.) 
EDIT: Here's the above, stated differently. Let $f: S^2 \to S^2$ be a map of nonzero degree. I claim you cannot extend $f$ to $\Bbb{CP}^2$. For $\Bbb{CP}^2$ is obtained from $S^2$ by attaching a 4-cell along the Hopf map $\eta: S^3 \to S^2$, it's only possible to extend the map over the 4-cell if $f\eta$ is null-homotopic; but $f\eta = n[\eta] \in \pi_3(S^2) \cong \Bbb Z$, where the Hopf map is a generator of this homotopy group.
The reason these are the same answer is because the way one often proves that $n\eta$ is not null-homotopic is by calculating the cohomology ring structure on $X_n = S^2 \cup_{n\eta} D^4$ and seeing that it's nontrivial.
