Equivariant Steenrod Operations

Abstract

In this paper, we introduce the concept of $\mathrm{R}$-Eulerian sequences for any $\mathcal{N}_\infty$-ring spectrum $\mathrm{R}$ with finite orientation order. We show that each $\mathrm{R}$-Eulerian sequence corresponds to a stable $\mathrm{R}$-cohomology operation. Moreover, we show that the collection of $\mathrm{R}$-Eulerian sequences admits an additive and a multiplicative structure that are linear over a coefficient ring. It remains an open question whether operations coming from Eulerian sequences generate the algebra of all stable $\mathrm{R}$-cohomology operations. Finally, we apply our theory to equivariant ordinary cohomology with coefficients in finite fields to produce genuine equivariant lifts of the classical Steenrod operations for all finite groups.