Given a symplectic form $\omega$ on a symplectic vector space $V$, a complex structure $J$ on $V$ is said to be tamed by $\omega$ or $\omega$- tamed if $$\omega(v,Jv)>0$$ for all non zero $v\in V$. Denote the space of all $\omega$-tamed complex structure on $V$ by $\mathcal{J}_{\tau}(V,\omega)$. I want to understand the proof of Gromov where he showed that $\mathcal{J}_{\tau}(V,\omega)$ is contractible. But there is one step in the poof where he just state it as obvious without justification. That step is the following:
Denote by $\Omega(V)$ the space of all symplectic forms on $V$ and denote by $\mathcal{J}(V)$ the space of all complex structures on $V$, and consider the subspace $C_{\tau}(V)=\{(\omega. J)\in \Omega(V)\times \mathcal{J}(V)~|~J \text{ is $\omega$-tamed}\}$ of $\Omega(V)\times\mathcal{J}(V)$. If we let $\pi:C_{\tau}(V)\to \Omega(V)$ to be the projection map onto the first factor, then Gromov showed that $\pi$ is a homotopy equivalence (this part I understood) but he also said that $\pi$ is also a fibration. This part I could not show how is $\pi$ a fibration. I know the definition of a fibration, but the difficulty is I dont have any clue how to construct a lift of the homotopy.