Classical phase space is defined as a space in which all possible states are represented. Every state corresponds to a unique point in the phase space.
On the other hand, in quantum mechanics every quantum mechanical observable corresponds to a unique function (distribution) on phase space.
In quantum mechanics we usually deal with Wigner function in $(x,p)$ space, but there also exist Glauber-Sudarshan $P$ or Husimi $Q$ distribution in parameter space of coherent states. This concept was firstly applied to Hamiltonians with Heisenberg-Weyl dynamical group, but other hamiltonians with different dynamical group (e.g. $SU(2)$) have formulation in phase space - parametric space of generalized coherent space (Perelomov).
In former case, both phase space $(x,p)$ and coherent state is flat - it is just a plane. In the concept of generalized coherent states for e.g. $SU(2)$ group, parameter space can be identified with sphere.
What is the topology of the associated phase space with Wigner function?
Can we only determine topology of the phase space, or the geometry is also possible?
Are all those formulations with Wigner, Glauber, Husimi etc. equivalent?