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Steenrod closed parameter ideals in the mod-$2$ cohomology of $A_4$ and $SO(3)$

Henrik Rüping, Marc Stephan, Ergun Yalcin

TL;DR

The paper develops a complete classification of Steenrod-closed parameter ideals in mod-$2$ cohomology rings tied to $A_4$ and $SO(3)$, recasting free actions on products of spheres in terms of $k$-invariants arising as kernels of classifying maps. It identifies three families—fibered, twisted, and mixed—of parameter ideals in $H^*(BA_4;\mathbb{F}_2)$ (and their Dickson algebra analog in $H^*(BSO(3);\mathbb{F}_2)$), with precise degree constraints and a uniqueness property for each degree pair. Using invariant theory, Kameko maps, and detailed Steenrod algebra calculations, it derives necessary obstructions on dimensions $(n,m)$ for free actions, showing these obstructions persist under restriction to $SO(3)$ and yield density-zero results in the large-degree limit. The results extend Oliver’s obstruction to $A_4$-actions, provide a structured framework for analyzing rank-2 finite groups, and yield explicit degree patterns that must be satisfied for any free action on $S^n\times S^m$.

Abstract

In this paper, we classify the parameter ideals in $H^*(BA_4;\mathbb{F}_2)$ and in the Dickson algebra $H^*(BSO(3);\mathbb{F}_2)$ that are closed under Steenrod operations. Consequently, we obtain restrictions on the dimensions $n,m$ for which $A_4$ (and $SO(3)$) can act freely on $S^n\times S^m$.

Steenrod closed parameter ideals in the mod-$2$ cohomology of $A_4$ and $SO(3)$

TL;DR

The paper develops a complete classification of Steenrod-closed parameter ideals in mod- cohomology rings tied to and , recasting free actions on products of spheres in terms of -invariants arising as kernels of classifying maps. It identifies three families—fibered, twisted, and mixed—of parameter ideals in (and their Dickson algebra analog in ), with precise degree constraints and a uniqueness property for each degree pair. Using invariant theory, Kameko maps, and detailed Steenrod algebra calculations, it derives necessary obstructions on dimensions for free actions, showing these obstructions persist under restriction to and yield density-zero results in the large-degree limit. The results extend Oliver’s obstruction to -actions, provide a structured framework for analyzing rank-2 finite groups, and yield explicit degree patterns that must be satisfied for any free action on .

Abstract

In this paper, we classify the parameter ideals in and in the Dickson algebra that are closed under Steenrod operations. Consequently, we obtain restrictions on the dimensions for which (and ) can act freely on .
Paper Structure (8 sections, 59 theorems, 143 equations, 1 figure)

This paper contains 8 sections, 59 theorems, 143 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Recollection of invariant theory
  3. Classification of twisted ideals
  4. Observations on nontwisted ideals
  5. Classification of fibered ideals
  6. Classification of Steenrod closed parameter ideals
  7. The k-invariants of a free G-action
  8. Table of Steenrod closed parameter ideals

Key Result

Theorem 1.1

Let $I\subset H^*(BA_4;\mathbb F_2)$ be a nonzero ideal generated by homogeneous elements of the same degree $i$. If $I$ is closed under the Steenrod operations, then $I=\langle v^k\rangle$ for $k=i/3$.

Figures (1)

  • Figure 1: Steenrod closed parameter ideals with parameters of degrees at most $60$.

Theorems & Definitions (121)

  • Theorem 1.1: Oliver
  • Theorem 1.2: \ref{['thm:classification_steenrodclosedparameterideals']}, \ref{['pro:uniquenessindegrees']}
  • Theorem 1.3: \ref{['thm:ObstructionskInvariants']}, \ref{['thm:ObstructionskInvariants_integral']}
  • Theorem 1.4: \ref{['cor:percentage']}
  • Theorem 2.1: adem2013cohomology
  • Definition 2.2
  • Lemma 2.3: see derksenkemper2015
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 111 more