I'm reading a passage in Milnor-Stasheff about Euler class, and I noticed that he states that the projection map $$\pi \colon E \to B $$ where $(E,\pi,B)$ is a n-dim vector bundle, induces a canonical isomorphism $$ \pi^* \colon H^n(B) \to H^n(E).$$
Clearly by the fact that we have $\pi\circ s_0 = Id_B$ we have that $\pi^*$ is injective. But I have no idea on how to prove surjectiveness.
Can someone help me?
The map $F: E\times [0,1]\to E$ defined fiberwise by $(b,x)\mapsto (b,tx)$ is continuous and is a deformation retraction from the identity on $E$ (in $t=1$) to $B$ (in $t=0$), where $B$ is identified with the zero section of $E$. A deformation retraction induces cohomology isomorphism.
(For the other direction, as the identity on $E$ and the projection to $B$ are homotopic, so $s_0 \circ \pi: E\to E$ is homotopic to $\mathrm{id}: E\to E$ and therefore $\pi^*$ is surjective.)