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arXiv · 2401.07919

Steenrod operations on polyhedral products

Sanjana Agarwal·Jelena Grbić·Michele Intermont·Milica Jovanović·Evgeniya Lagoda·Sarah Whitehouse
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Abstract

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.

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Sanjana Agarwal, Jelena Grbić, Michele Intermont, Milica Jovanović, Evgeniya Lagoda, Sarah Whitehouse. 2024-01-15. Steenrod operations on polyhedral products. https://arxiv.org/abs/2401.07919

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