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  Let M be a manifold with a nondegenetate 2-form \(\omega\). Let \(\mathcal{J}\) denote the set of \(\omega\)-tamed almost complex structures. In other words, an almost complex structure J belongs to \(\mathcal{J}\) if \(\omega(X,JX)>0\) for all nonzero \(X\in TM\). I don’t see why this space is contractile. I can show that the space of \(\omega\)-tamed almost complex structures on a symplectic vector space is contractible but this argument depends on a choice of symplectic basis. This ar...