# Injectivity on rational homtopy implies surjectivity on rational cohomology for classifiying spaces BO and BTOP

Take $$BO=\bigcup_{n\geq 1}BO(n)$$ and $$BTOP=\bigcup_{n\geq 1}BTOP(n)$$ where $$TOP(n)$$ is the set of homeomorphisms of $$\mathbb{R}^n$$ which send $$0$$ to $$0$$ and let $$\phi: BO \rightarrow BTOP$$ be the map induced by the inclusion of $$O(n) \rightarrow TOP(n)$$.
I've already shown that $$\phi_{*}: \pi_i(BO) \otimes \mathbb{Q} \rightarrow \pi_i(BTOP) \otimes \mathbb{Q}$$ is injective for $$i\geq 0$$ and I want to show that this implies $$\phi^*: H^i(BTOP;\mathbb{Q})\rightarrow H^i(BO;\mathbb{Q})$$ surjective for $$i\geq 0$$.
I first tried to use the rational Huerwics and the fact that $$H^*(BO;\mathbb{Q}) \cong \mathbb{Q}[p_1,p_2,\dots]$$ where $$p_i$$ are the universal Pontryagin classes of the $$EO \rightarrow BO$$ bundle, but it seems that this is not enough. Then I thought about that $$BO$$, $$BTOP$$ are H-spaces and $$\phi$$ is a H-map and may work with Hopf-algebras, but the cohomology of $$BTOP$$ is not finally generated, at least I don't see why it should be, so $$H^*(BTOP,\mathbb{Q})$$ doesn't allow a Hopf-algebra structure.
I would be grateful for any advice or hint.

Write $$kO$$ for the connective $$k$$-theory, and $$X$$ for the connective delooping of $$BTOP$$. Then $$H_*(BO;Q)$$ and $$H_*(BTOP,Q)$$ are free commutative (in graded sense) generated by $$\pi _*(kO,Q)\cong \pi _*(BO,Q)$$ and $$\pi _*(X,Q)\cong \pi _* (BTOP,Q)$$ respectively. Thus the injection of the homotopy groups imply the injection of the homology. By dualizing ($$H_*(BO,Q)$$ is surely of finite type, and this suffices) you get the surjection of homotopy groups.