If $(X,A)$ is 8-connected cw pairs, why $H^7(X;\mathbb{Z}) \to H^7(A;\mathbb{Z})$ is onto?
This is my question of calculating the cohomology rings of unitary groups using Leray-Hirsch theorem. Since $U(n-1)\to U(n) \to S^{2n-1}$ is a fibre bundle, $(U(n),U(n-1))$ is $(2n-2)$ connected from the long exact homotopy sequence. Hatcher says it is a surjection of the $i$-th cohomology for all $i\le (2n-3)$ then the condition of the Leray-Hirch theorem satisfied. This confused me a lot.
I realized that connected pairs is defined by the first nontrivial relative homotopy group. And we have the hurewicz theorem which says the first nonzero homotopy group is isomorphic to the first nonzero homology group. But these sounds nothing help to solve my problem because it is cohomology but not homology. I try to use the definition of cohomology to prove the surjection. However, I can't figure it out. Any help is appreciated. Thanks.
By the universal coefficient theorem, if $H_n(X,A)$ and $H_{n-1}(X,A)$ are both trivial, then $H^n(X,A)$ is trivial as well. So if $H_n(X,A)$ is trivial for all $n\leq 8$, then $H^n(X,A)$ is trivial for all $n\leq 8$ as well.
Alternatively, any $n$-connected pair $(X,A)$ is weak homotopy equivalent to a CW pair $(Y,B)$ where $Y\setminus B$ contains no cells of degree $\leq n$. The cellular cochain complex for computing $H^*(Y,B)$ is then trivial in degrees $\leq n$.