TITLE: Triviality of a fiber bundle
QUESTION [5 upvotes]: Is the principal fiber bundle $GL^+(6,\mathbb R)$ over $GL^+(6,\mathbb R)/SL(3,\mathbb C)$ trivial ?

REPLY [6 votes]: No. One can reduce to the maximal compact subgroups. So the question is, if $U(3)\rightarrow SO(6)\rightarrow SO(6)/U(3)$ is trivial. But $SO(6)/U(3)\cong \mathbb{C}P^3$ (this can be seen by using $Spin(6)\cong SU(4)$, for instance). But $\pi_2(U(3)\times \mathbb{C}P^3) = \pi_2(\mathbb{C}P^3) \cong \mathbb{Z}$ and $\pi_2(SO(6)) \cong \pi_2(SO(3))\cong \pi_2(S^3) = 0$.