Steenrod closed $C_3$-invariant parameter ideals in the mod 2 cohomology of $\mathbb{Z}/2\times\mathbb{Z}/2$
Henrik Rüping, Marc Stephan
TL;DR
The paper addresses the problem of classifying $C_3$-invariant parameter ideals in $H^*(BP)\cong \mathbb{F}_2[a,b]$ that are closed under Steenrod operations, connecting these algebraic invariants to topological $A_4$-actions on finite CW complexes with four-dimensional mod $2$ cohomology. It shows that the remaining Steenrod-closed $C_3$-invariant parameter ideals arise precisely from the $C_3$-orbits of $a^n$ with $n$ a power of two, yielding the ideals $\langle a^n,b^n\rangle$, while ideals extending from $H^*(BA_4)$ are treated via the established correspondence. The authors develop criteria to distinguish extension-from-$H^*(BA_4)$ ideals from orbit-generated ones, employing Steenrod operations and Kameko maps, and prove that the only orbit-generated Steenrod-closed cases occur for $n=2^t$. Topologically, these results constrain possible $P$-free $A_4$-actions on four-dimensional cohomology complexes (e.g., realizing $n\in\{0,1,3,7\}$ for products of spheres) and establish rigidity of the corresponding cellular cochain complexes, showing the annihilator ideals in $H^*(BP)$ determine the homotopy type of the action.
Abstract
For the nontrivial action by the cyclic group $C_3$ of order $3$ on the graded polynomial ring $\mathbb{F}_2[a,b]$, we classify the $C_3$-invariant parameter ideals that are closed under Steenrod operations. The classification has applications to free actions by the Klein four-group $\mathbb{Z}/2\times\mathbb{Z}/2$ on products of two spheres (and more generally, finite CW complexes with four-dimensional mod $2$ homology) that extend to actions by the alternating group $A_4=(\mathbb{Z}/2\times\mathbb{Z}/2)\rtimes C_3$.