Question

I'd like to build a model for the space $EU(n)$: the total space of universal bundle $\pi:EU(n) \rightarrow BU(n)$. $\;$ $EU(n)$ must be a conctractible space on which $U(n)$ acts freely. So I consider Stiefel manifold $V_{n}(\mathbb{C}^{k})$ of $\;$ $n-$frame in $\mathbb{C}^{k}$ and the associated fiber bundle $$ V_{n-1}(\mathbb{C}^{k-1}) \rightarrow V_{n}(C^{k}) \rightarrow S^{2k-1} $$ The induced sequence in homotopy shows that all homotopy gruops vanish for $k$ large. Indeed, the natural action of $U(n)$ on $V_{n}(\mathbb{C}^{k})$ is free (the quotien is the Grassmannian manifold). So I can choose $$ EU(n)=\lim_{k \to \infty} V_{n}(\mathbb{C}^{k}) = V_{n}(\mathbb{C}^{\infty}) $$ the direct limit of $V_{n}(\mathbb{C}^{k})$. How could I prove that the action of $U(n)$ on $V_{n}(\mathbb{C}^{k})$ is still free? In other words, why is the limit of a free action free?

Answer 1

See section 47 of (here).