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Let $T$ be a torus of dimension $n>1$ and $M$ a compact $T-$manifold. $M$ is a GKM manifold if the set of zero dimensional orbits in the orbit space $M/T$ is zero dimensional and the set of one dimensional orbits in $M/T$ is one…

For an equivariantly formal action of a compact torus $T$ on a smooth manifold $X$ with isolated fixed points we investigate the global homological properties of the graded poset $S(X)$ of face submanifolds. We prove that the condition of…

Let T be a torus of dimension at least k and M a T-manifold. M is a GKM_k-manifold if the action is equivariantly formal, has only isolated fixed points, and any k weights of the isotropy representation in the fixed points are linearly…

An algebraic GKM manifold is a non-singular algebraic variety equipped with an algebraic action of an algebraic torus, with only finitely many torus fixed points and finitely many 1-dimensional orbits. In this expository article, we use…

A smooth GKM stack is a smooth Deligne-Mumford stack equipped with an action of an algebraic torus $T$, with only finitely many zero-dimensional and one-dimensional orbits. (i) We define the stacky GKM graph of a smooth GKM stack, under the…

Let $M^{2d}$ be a compact symplectic manifold and $T$ a compact $n$-dimensional torus. A Hamiltonian action, $\tau$, of $T$ on $M$ is a GKM action if, for every $p \in M^T$, the isotropy representation of $T$ on $T_pM$ has pair-wise…

We consider a Hamiltonian action of n-dimensional torus, T^n, on a compact symplectic manifold (M,\omega) with d isolated fixed points. For every fixed point p there exists (though not unique) a class a_p in H^*_{T}(M; Q) such that the…

We show that for a Hamiltonian action of a compact torus $G$ on a compact, connected symplectic manifold $M$, the $G$-equivariant cohomology is determined by the residual $S^1$ action on the submanifolds of $M$ fixed by codimension-1 tori.…

We calculate the integral equivariant cohomology, in terms of generators and relations, of locally standard torus orbifolds whose odd degree ordinary cohomology vanishes. We begin by studying GKM-orbifolds, which are more general, before…

We give a characterization of locally standard, $\mathbb{Z}$-equivariantly formal manifolds in general position. In particular, we show that for dimension $2n$ at least $10$, to every such manifold with labeled GKM graph $\Gamma$ there is…

Let $T$ be a torus and $B$ a compact $T-$manifold. Goresky, Kottwitz, and MacPherson show in \cite{GKM} that if $B$ is (what was subsequently called) a GKM manifold, then there exists a simple combinatorial description of the equivariant…

We prove for closed, odd-dimensional GKM$_3$ manifolds of non-negative sectional curvature that both the equivariant and the ordinary rational cohomology split off the cohomology of an odd-dimensional sphere.

In this paper we study non-negatively curved and rationally elliptic GKM$_4$ manifolds and orbifolds. We show that their rational cohomology rings are isomorphic to the rational cohomology of certain model orbifolds. These models are…

In this paper, we study Hamiltonian R-actions on symplectic orbifolds [M/S], where R and S are tori. We prove an injectivity theorem and generalize Tolman-Weitsman's proof of the GKM theorem in this setting. The main example is the…

We study Cohen-Macaulay actions, a class of torus actions on manifolds, possibly without fixed points, which generalizes and has analogous properties as equivariantly formal actions. Their equivariant cohomology algebras are computable in…

We classify $T^2$-GKM fibrations in which both fiber and base are the GKM graph of $S^4$, with standard weights in the base. For each case in which the total space is orientable, we construct, by explicit clutching, a realization as a…

We establish a localization theorem of Borel-Atiyah-Segal type for the equivariant operational K-theory of Anderson and Payne. Inspired by the work of Chang-Skjelbred and Goresky-Kottwitz-MacPherson, we establish a general form of GKM…

The Chang-Skjelbred method computes the cohomology of a suitable space with a torus action from its equivariant one-skeleton. We show that, under certain restrictions on the cohomological torsion, the integral cohomology is encoded in the…

Let T be a compact torus and X a nice compact T-space (say a manifold or variety). We introduce a functor assigning to X a "GKM-sheaf" F_X over a "GKM-hypergraph" G_X. Under the condition that X is equivariantly formal, the ring of global…

In this paper we first consider the Hamiltonian action of a compact connected Lie group on an $H$-twisted generalized complex manifold $M$. Given such an action, we define generalized equivariant cohomology and generalized equivariant…