Local and global combinatorial structure of torus actions

Consider a smooth torus action on a connected closed manifold $X$ of real dimension $2n$ such that the fixed point set is finite and nonempty. The action is called equivariantly formal if odd degree cohomology of X vanishes. For example, hamiltonian actions on compact manifolds with isolated fixed points are equivariantly formal.
Connected components of $X^G$, for some subgroup $G$ of $T$, are called face submanifolds of $X$. We study the graded poset $S_X$ of face submanifolds. The motivating and well studied examples are actions of comlexity zero (torus manifolds). In these examples $S_X$ (with reversed order) is a simplicial poset - this is a general local property. If $X$ is equivariantly formal, $S_X$ is a triangulated sphere - this is a global property.
In my talk I want to show how these facts generalize to positive complexity. The talk is based on several works (as well as works in progress) with Mikiya Masuda, Grigory Solomadin and Vladislav Cherepanov.