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

TL;DR

This paper classifies Steenrod-closed parameter ideals in the mod-2 cohomology of A_4 and SO(3), deriving restrictions on their free actions on products of spheres.

Contribution

It provides a classification of Steenrod-closed parameter ideals in specific cohomology rings, revealing new constraints on group actions on spheres.

Findings

Classified Steenrod-closed parameter ideals in H^*(BA_4;F_2) and H^*(BSO(3);F_2)

Derived restrictions on dimensions for free actions of A_4 and SO(3) on S^n×S^m

Enhanced understanding of the structure of cohomology rings under Steenrod operations

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 $S^n\times S^m$.