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GKM manifolds are not rigid

### Oliver Goertsches, Panagiotis Konstantis and Leopold Zoller

Algebraic & Geometric Topology 22 (2022) 3511–3532
##### Abstract

We construct effective GKM ${T}^{3}$–actions with connected stabilizers on the total spaces of the two ${S}^{2}$–bundles over ${S}^{6}$ with identical GKM graphs. This shows that the GKM graph of a simply connected integer GKM manifold with connected stabilizers does not determine its homotopy type. We complement this by a discussion of the minimality of this example: the homotopy type of integer GKM manifolds with connected stabilizers is indeed encoded in the GKM graph for smaller dimensions, lower complexity, or lower number of fixed points. Regarding geometric structures on the new example, we find an almost complex structure which is invariant under the action of a subtorus. In addition to the minimal example, we provide an analogous example where the torus actions are Hamiltonian, which disproves symplectic cohomological rigidity for Hamiltonian integer GKM manifolds.

##### Keywords
GKM theory, Hamiltonian action, equivariant cohomology, cohomological rigidity
##### Mathematical Subject Classification
Primary: 57R91
Secondary: 53D20, 55N91