Assume $M$ is a closed smooth manifold of real dimension $\geq 4$. What is known about the geometry of the "space" of almost complex structures up to conjugation by diffeomorphisms? There are quotes because I am not sure what is the natural topology to consider on that set. Is it locally finite-dimensional? Is there a cohomological interpretation of the tangent space at a point $[J]$ (corresponding to an a.c.s. $J$)?
