On torus equivariant $S^4$-bundles over $S^4$ and Petrie-type questions for GKM manifolds

Oliver Goertsches; Panagiotis Konstantis; Leopold Zoller

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arXiv · 2509.03100

On torus equivariant $S^4$-bundles over $S^4$ and Petrie-type questions for GKM manifolds

Oliver Goertsches·Panagiotis Konstantis·Leopold Zoller

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Abstract

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 $T^2$-equivariant linear $S^4$-bundle over $S^4$. 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.

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BibTeX RIS

Oliver Goertsches, Panagiotis Konstantis, Leopold Zoller. 2025-09-03. On torus equivariant $S^4$-bundles over $S^4$ and Petrie-type questions for GKM manifolds. https://arxiv.org/abs/2509.03100

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