the universal cover of the real(or complex) Grassimannian

Question

As we all know that $S^n$ is the universal cover of $\mathbb{R}P^n=Gr_1(n+1)$, my question is:

Is there a description of the universal cover of $Gr_k(n)$, the Grassmann complex consisting of $k$-planes in $\mathbb{R}^n$? Concrete examples are also welcome.

Thank you!

Answer 1

The universal cover of the real grassmannian is a double cover called the oriented grassmannian, points are a $k$-plane plus a choice of orientation. See Fundamental groups of Grassmann and Stiefel manifolds.