Equivariant Steenrod Operations

TL;DR

This paper develops a new framework for equivariant Steenrod operations using $ ext{R}$-Eulerian sequences in the context of $ ext{R}$-ring spectra, leading to the construction of genuine equivariant Steenrod operations for finite groups.

Contribution

It introduces $ ext{R}$-Eulerian sequences for $ ext{R}$-ring spectra and constructs equivariant Steenrod operations for finite groups, expanding the algebraic tools in equivariant cohomology.

Findings

Defined $ ext{R}$-Eulerian sequences for $ ext{R}$-ring spectra.

Established a stable $ ext{R}$-cohomology operation from these sequences.

Constructed genuine equivariant Steenrod operations for all finite groups.

Abstract

We introduce the notion of $\mathrm{R}$-Eulerian sequences for any $\mathcal{N}_\infty$-ring spectrum $\mathrm{R}$ of finite orientation order. We prove that each $\mathrm{R}$-Eulerian sequence determines a stable $\mathrm{R}$-cohomology operation. Furthermore, we show that the collection of $\mathrm{R}$-Eulerian sequences carries a natural additive and a multiplicative structure which is linear over the coefficient ring. As an application, we specialize to equivariant ordinary cohomology with coefficients in finite fields and construct genuine equivariant Steenrod operations for all finite groups.