Steenrod operations on polyhedral products

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2025-05-27 DOI:10.1016/j.topol.2025.109446

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

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引用次数: 0

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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多面体积的Steenrod运算

讨论了mod2 Steenrod代数对各种多面体积和相关空间上同调的作用。我们对Davis-Januszkiewicz空间及其推广、矩角复合体以及某些多面体连接进行了验证。通过研究底层简单复合体的组合学，我们推导出了可以出现非平凡Steenrod运算的最低上同维数的一些结果。给出了简单复合体上Steenrod运算的链级公式。我们解释了将这样的公式从简单复K“传播”到K上的多面体连接的思想，并给出了这个过程的例子。我们将Steenrod代数作用在多面体连接上的传播与矩角复合体上的传播联系起来。虽然在这些情况下，人们可以通过稳定同伦分解来理解Steenrod作用，但我们期望将这种方法应用于没有这种分解的情况。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.