Classifying space of $GL_{n}(\mathbb{F})$?

Question

I was looking for the classifying space of the general linear group $GL_{n}(\mathbb{F})$ over a field (of characteristic either zero or positive, finite or infinite), but unfortunately I didn't manage to find something. Can you help me please? Also, do you know what's the cohomology ring of that group in the case where the field is finite or infinite?

Answer 1

I think that I can give an answer, at least for the very first part of my question. Assume that $GL_{n}(\mathbb{R})$ is the general linear over $\mathbb{R}$. It works almost in the same fashion for $\mathbb{C}$ as well. Then, someone can prove that the "natural map" $\rho:V_{n}{\mathbb{R}^{\infty}} \rightarrow G_{n}{\mathbb{R}^{\infty}}$ from the $n$-frames into the infinite dimesional real vector space to the $n$-dimensional Grassmanian of the same space is a $GL_{n}(\mathbb{R})$-bundle. Also the total space is contractible, so the Grassmanian is a model for $BGL_{n}(\mathbb{R})$. If we restrict our attention into the case where our total space is a Stiefel manifold of the $\textbf{orthonormal}$ $n$-frames, which I will denote by ${V^{O}_{n}} {\mathbb{R}^{\infty}}$ for simplicity, because of the Gram-Schmidt Theorem we end up by a deformation retract between the Stiefel manifold and $V_{n}{\mathbb{R}^{\infty}}$. So that gives an $O_{n}(\mathbb{R})$-bundle, $\tilde{\rho}: {V^{O}_{n}} {\mathbb{R}^{\infty}} \rightarrow G_{n}{\mathbb{R}^{\infty}}$. Hence $BGL_{n}(\mathbb{R})=BO_{n}(\mathbb{R})$.

Any comments, or corrections are really welcome!

Also any comment for cohomology of $GL_{n}(\mathbb{F})$ for any kind of field may be helpful!