So let $$F_n(\mathbb{C}^n)\overset{i}{\rightarrow}F_n(\mathbb{C}^\infty)\overset{p}{\rightarrow}G_n(\mathbb{C}^\infty)$$
Where $F_n$ are $n$-tuples of orthogonal vectors and $G_n$ is the Grassmanian.Note that $p$ is the map which takes the $n$-tuple of orthogonal vectors to the subspace it spans.
So here is my question. Hatcher doesn't make any further remarks, but why is $\text{ker}i^\ast\subseteq\text{im}p^\ast$, where we consider the induced maps on the cohomology rings.