Is $GL(n,\mathbb{C})$ contractible for any $n$?
My intuition is telling me it is not, because the determinant maps the general linear to $\mathbb{C}\setminus 0$ which is not contractible. If there was an homotopy of a cycle in $GL_n$ to a point, the determinant would map this to an analogous homotopy in $\mathbb{C}\setminus 0$. Is there any missing subtlety to this argument?
My real question is: how do write the complex general linear group as the union of finitely many contractible patches?
I am interested in explicit descriptions of the covering.
EDIT: The problem has been reduced (in the comments) to covering $U(n)$.