Steenrod operations on polyhedral products
Fuente:
arXiv
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| Autores principales: | , , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866909227528749056 |
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| author | Agarwal, Sanjana Grbić, Jelena Intermont, Michele Jovanović, Milica Lagoda, Evgeniya Whitehouse, Sarah |
| author_facet | Agarwal, Sanjana Grbić, Jelena Intermont, Michele Jovanović, Milica Lagoda, Evgeniya Whitehouse, Sarah |
| contents | We describe the action of the mod $2$ Steenrod algebra on the cohomology of various polyhedral products and related spaces. We carry this out for Davis-Januszkiewicz spaces and their generalizations, for moment-angle complexes as well as for certain polyhedral joins. By studying the combinatorics of underlying simplicial complexes, we deduce some consequences for the lowest cohomological dimension in which non-trivial Steenrod operations can appear.
We present a version of cochain-level formulas for Steenrod operations on simplicial complexes. We explain the idea of "propagating" such formulas from a simplicial complex $K$ to polyhedral joins over $K$ and we give examples of this process. We tie the propagation of the Steenrod algebra actions on polyhedral joins to those on moment-angle complexes. Although these are cases where one can understand the Steenrod action via a stable homotopy decomposition, we anticipate applying this method to cases where there is no such decomposition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07919 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Steenrod operations on polyhedral products Agarwal, Sanjana Grbić, Jelena Intermont, Michele Jovanović, Milica Lagoda, Evgeniya Whitehouse, Sarah Algebraic Topology 55U10, 05E45, 13F55, 55S10 We describe the action of the mod $2$ Steenrod algebra on the cohomology of various polyhedral products and related spaces. We carry this out for Davis-Januszkiewicz spaces and their generalizations, for moment-angle complexes as well as for certain polyhedral joins. By studying the combinatorics of underlying simplicial complexes, we deduce some consequences for the lowest cohomological dimension in which non-trivial Steenrod operations can appear. We present a version of cochain-level formulas for Steenrod operations on simplicial complexes. We explain the idea of "propagating" such formulas from a simplicial complex $K$ to polyhedral joins over $K$ and we give examples of this process. We tie the propagation of the Steenrod algebra actions on polyhedral joins to those on moment-angle complexes. Although these are cases where one can understand the Steenrod action via a stable homotopy decomposition, we anticipate applying this method to cases where there is no such decomposition. |
| title | Steenrod operations on polyhedral products |
| topic | Algebraic Topology 55U10, 05E45, 13F55, 55S10 |
| url | https://arxiv.org/abs/2401.07919 |