On torus equivariant $S^4$-bundles over $S^4$ and Petrie-type questions for GKM manifolds
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866911167687950336 |
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| author | Goertsches, Oliver Konstantis, Panagiotis Zoller, Leopold |
| author_facet | Goertsches, Oliver Konstantis, Panagiotis Zoller, Leopold |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_03100 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On torus equivariant $S^4$-bundles over $S^4$ and Petrie-type questions for GKM manifolds Goertsches, Oliver Konstantis, Panagiotis Zoller, Leopold Algebraic Topology 57R91, 55N91 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. |
| title | On torus equivariant $S^4$-bundles over $S^4$ and Petrie-type questions for GKM manifolds |
| topic | Algebraic Topology 57R91, 55N91 |
| url | https://arxiv.org/abs/2509.03100 |