GKM manifolds are not rigid

TL;DR

This paper constructs examples of GKM manifolds with identical GKM graphs but different homotopy types, demonstrating that GKM graphs do not always determine the manifold's homotopy type, especially in higher dimensions.

Contribution

It provides explicit examples of GKM manifolds with the same GKM graph but different homotopy types, challenging the rigidity conjecture in GKM theory.

Findings

GKM graphs do not determine homotopy type in certain cases

Constructed examples of non-rigid GKM manifolds with identical graphs

Disproved symplectic cohomological rigidity for Hamiltonian GKM manifolds

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.