On torus equivariant S⁴-bundles over S⁴ and Petrie-type questions for GKM manifolds

2025/09/03 by Oliver Goertsches, Goertsches, Oliver, Panagiotis Konstantis +3
Mathematics · #55N91 #57R91 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2509.03100

openalex publication_date 2025/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We classify T²-GKM fibrations in which both fiber and base are the GKM graph of S⁴, 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 T²-equivariant linear S⁴-bundle over S⁴. We determine which of the total spaces of these examples are non-equivariantly homotopy equivalent, homeomorphic or diffeomorphic, thereby finding many examples of a) pairs of homotopy equivalent, non-homeomorphic GKM manifolds with different first Pontryagin class, and b) pairs of GKM actions on the same smooth manifold whose GKM graphs do not agree as unlabeled graphs.

Citations

• On GKM fiber bundles and realizability with full flag fibers
• On the Stiefel-Whitney classes of GKM manifolds
• Low-dimensional GKM theory
• The GKM correspondence in dimension 6
• One-skeleta, Betti numbers and equivariant cohomology

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• Realization of GKM fibrations and new examples of Hamiltonian non-Kähler actions
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• Classification of equivariant vector bundles over two-torus