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Stable Adams operations on $RO(C_2)$-graded homotopy groups

Anton Engelmann

TL;DR

The paper determines the C2-equivariant stable Adams operations on the RO(C2)-graded homotopy of KR and TMF1(3) by exploiting E∞-ring endomorphisms and the strong-even property, together with étale descent arguments. The main result expresses ψ^k on π^{C2}_{a+bσ} as multiplication by k^{(a+b)/2} after inverting k, with 2-torsion fixed, and explicit checks are performed on generators in both KR and TMF1(3). The approach reduces to known KO and TMF0(3) computations via restriction maps and the Davies-Hecke construction of Adams operations. These findings provide explicit C2-equivariant Adams operations for KR and TMF1(3), clarifying the link between equivariant KO/KR and level structure TMF theories.

Abstract

The $C_2$-spectrum of Atiyah's Real $K$-theory is denoted by $\mathbf{KR}$ and the $C_2$-spectrum of topological modular forms of level structure $Γ_1(3)$ by $\mathbf{TMF}_1(3)$. In this short note we compute the $C_2$-equivariant stable Adams operations on the $RO(C_2)$-graded homotopy groups of $\mathbf{KR}$ and $\mathbf{TMF}_1(3)$.

Stable Adams operations on $RO(C_2)$-graded homotopy groups

TL;DR

The paper determines the C2-equivariant stable Adams operations on the RO(C2)-graded homotopy of KR and TMF1(3) by exploiting E∞-ring endomorphisms and the strong-even property, together with étale descent arguments. The main result expresses ψ^k on π^{C2}_{a+bσ} as multiplication by k^{(a+b)/2} after inverting k, with 2-torsion fixed, and explicit checks are performed on generators in both KR and TMF1(3). The approach reduces to known KO and TMF0(3) computations via restriction maps and the Davies-Hecke construction of Adams operations. These findings provide explicit C2-equivariant Adams operations for KR and TMF1(3), clarifying the link between equivariant KO/KR and level structure TMF theories.

Abstract

The -spectrum of Atiyah's Real -theory is denoted by and the -spectrum of topological modular forms of level structure by . In this short note we compute the -equivariant stable Adams operations on the -graded homotopy groups of and .
Paper Structure (3 sections, 4 theorems, 11 equations, 2 figures)

This paper contains 3 sections, 4 theorems, 11 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Recollections
  3. Computations of Adams operations

Key Result

Theorem A

Let $k \in {\mathbb{Z}}$ and $x \in \pi_{a+b\sigma}^{C_2} \mathbf{KR}[\frac{1}{k}]$, where $\sigma$ is the sign representation of $C_2$. Then In particular, if $x$ is torsion (necessarily $2$-torsion), then $\psi^k(x)=x$.

Figures (2)

  • Figure 1: Black circles, squares and lines represent copies of ${\mathbb{Z}}$, red dots and lines represent copies of $\mathbb{F}_{2}$. For more details we refer to Greenlees:4reality.
  • Figure 2: Black circles, squares and lines represent copies of ${\mathbb{Z}}$; red dots represent copies of $\mathbb{F}_{2}$, red lines mean a copy of $\mathbb{F}_{2}[\overline{a}_1, \overline{a}_3]$ and green lines mean a copy of $\mathbb{F}_{2}[\overline{a}_3]$. For more details we refer to GreenleesMeier:Gorenstein. The axes are the same as in \ref{['figure:preleiminaries:homotopyKR']}

Theorems & Definitions (7)

  • Theorem A: \ref{['thm:KR:main-thm']}
  • Theorem B: \ref{['thm:TMF:main-thm']}
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3