I have the following question: Let $M$ be an even dimensional Riemannian manifold. Under which conditions does there exists a homotopy to some symplectic manifold? is there any chance that such a homotopy exists even if $M$ is not symplectic? how does the homotopy look like? is it differentiable, only continous ... ? is there any chance that $M$ is homotopic to a complex manifold? Is there any reference in this direction ?

greetings mirta