Assume $m>n$. Consider projection map $\pi: C^m\to C^{m-1}$ by $(z_1,\dots, z_m)\to (z_1,\dots, z_{m-1})$. Let $G_n(V)$ be the set of codimension $n$ hyperplanes of $V$ and endow the topology as a subspace of projective space.
"This map induces the following continuous map $G_n(C^{m-1})\to G_n(C^m)$."
$\textbf{Q:}$ How so? I do not see why projection should induce such an obvious map. There is embedding $C^{m-1}\subset C^m$ by setting $z_m=0$. Then I have $G_n(C^{m-1})\to G_n(C^m)$ induced by this inclusion map by sending wedge product to wedge products where I have identified the image in the corresponding projective spaces.
Ref. Atiyah, K-theory Chpt 1 pg 28.
Writing $e_m=(0,\cdots,0,1)\in \mathbb C^m$, the required map is $$G_n(\mathbb C^{m-1})\to G_n(\mathbb C^m):L\mapsto L\oplus \mathbb Ce_m=\pi^{-1}(L)$$