Using the classification of 6-dimensional manifolds by Wall, Jupp andŽubr, we observe that the diffeomorphism type of simply-connected, compact 6-dimensional integer GKM T 2 -manifolds is encoded in their GKM graph. As an application, we show that the 6-dimensional manifolds on which Tolman and Woodward constructed Hamiltonian, non-Kähler T 2 -actions with finite fixed point set are both diffeomorphic to Eschenburg's twisted flag manifold SU(3)//T 2 . In particular, they admit a noninvariant Kähler structure.

We construct symplectic structures on roughly half of all equal rank biquotients of the form G//T , where G is a compact simple Lie group and T a torus, and investigate Hamiltonian Lie group actions on them. For the Eschenburg flag, this action has similar properties as Tolman's and Woodward's examples of Hamiltonian non-Kähler actions. In addition to the previously known Kähler structure on the Eschenburg flag, we find another Kähler structure on a biquotient SU(4)//T 3 .

This is a survey on the equivariant cohomology of Lie group actions on manifolds, from the point of view of de Rham theory. Emphasis is put on the notion of equivariant formality, as well as on applications to ordinary cohomology and to fixed points.

We classify fibrations of abstract 3-regular GKM graphs over 2-regular ones, and show that all fiberwise signed fibrations of this type are realized as the projectivization of equivariant complex rank 2 vector bundles over quasitoric 4-folds or S 4 . We investigate the existence of invariant (stable) almost complex, symplectic, and Kähler structures on the total space. In this way we obtain infinitely many Kähler manifolds with Hamiltonian non-Kähler actions in dimension 6 with prescribed one-skeleton, in particular with prescribed number of isolated fixed points.

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