4
$\begingroup$

I have the following question, which I dont really know if its true: Let $g : X \rightarrow Y$ be a continous map between two closed, oriented $n-$dimensional manifolds such that $g^{*} : H^{n}(Y, \mathbb{Q}) \rightarrow H^{n}(X, \mathbb{Q})$ is non-zero (here in this case we have $H^{n}(X, \mathbb{Q}) = \mathbb{Q}$ and $H^{n}(Y, \mathbb{Q}) = \mathbb{Q}$). How can one show that the map $g^{*} : H^{k}(Y, \mathbb{Q}) \rightarrow H^{k}(X, \mathbb{Q})$ is injective for $k ? Is this result true? If yes ow can one show it? thanks in advance!

ronald

  • 0
    Aahhh! I've made a huge mistake. I posted a "counterexample" using the Hopf manifold with its map to $\mathbb P^n$, but forgot that the dimensions of the manifolds don't match. Sorry about that.2012-05-29
  • 0
    does someone have any ideas ?2012-05-29
  • 0
    What have you tried? If you see closed, oriented manifold then you should probably think about Poincare duality. Then to get back cohomology information, you have the universal coefficient theorem. So follow your nose...it might all work!2012-05-29
  • 0
    well ok. this actually is equivalent to showing that $g_{*}: H_{k}(X, \mathbb{Q}) \rightarrow H_{k}(Y, \mathbb{Q})$ is surjective. But how to do that? I dont see any reason why this should be surjective. Do you ?2012-05-29
  • 0
    @ronald: Dear ronald, Your reformulation still hasn't taken into account Poincare duality, which is why it doesn't get you any further. Regards,2012-05-30

1 Answers 1