Steenrod operations on polyhedral products

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.

21 pages, v2 minor changes, accepted version to appear in special issue of Topology and its Applications dedicated to proceedings of the Women in Topology 4 workshop