Abstract. Interesting invariants of manifolds with compact, effective, smooth torus actions (with nonempty and finite fixed points) are recorded in the respective equivariant cohomology ring. A graph-theoretical approach to the cohomology description in a wide class of GKM manifolds is given by the Chang-Skjelbred sequence. However, the resulting answer is not very convenient, because it lacks generators and relations description. This was resolved with a face ring isomorphism by M. Masuda and T. Panov in complexity n-k=0 case. The case n-k=1 was partially resolved by A. Ayzenberg, M. Masuda and myself in 2022. In particular, for GKM T^k-actions on M^{2n} with n-k>0 the problem is open, in general. In this talk I will describe a new solution to the generators and relations problem in Hamiltonian, as well as in n-k<4 cases (under some additional technical assumptions). If time permits, I will outline a construction of new GKM 8-manifolds of complexity 0 whose orbit spaces bound Brieskorn homology 3-spheres (and do not satisfy the above assumptions). This follows by producing a smooth manifold with corners structure on the respective collars (the "disks, beams, plates" construction for the smooth conical stratification on the boundary), then using the recent realization theorem of Y. Karshon and S. Kuroki. Based on arXiv 2509.00392, 2604.13629, 2608.30513.