A description of the map between Grassmanians $G_1^k \rightarrow G_k$,

Question

We know that $G_k:=co\lim G_k(\Bbb C^n)$ is the classifying space for $k$ dimensional complex vector bundles. With total space $E_k = \{(x,v) \, :|, x \in G_k, v \in \Bbb C ^\infty \}$. So we may construct the $k$-vector bundle $(E_1)^k \rightarrow (G_1)^k$. By universality there should exists a map upto homotopy

$$(G_1)^k \rightarrow G_k $$

What is a description of this map?

I thought it would be simple, i.e. sending $k$ one-dimensional spaces to the $k$ -dimensional space they span - but this doesn't work? Since they need not be independent.

Answer 1

How about this: You can write $\mathbb{C}^\infty=(\mathbb{C}^\infty)^k$. Then each line can be seen in a different $\mathbb{C}^\infty$ and are therefore never linearly dependent. You take the span of $k$ lines to obtain the $k$-plane.