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$C_3$-equivariant stable stems

Yueshi Hou, Shangjie Zhang

TL;DR

This work delivers the first detailed computation of the 3-primary spoke-graded C3-equivariant stable stems π^{C3}_{i,j} for i ≤ 25 and −16 ≤ j ≤ 16, introducing spoke-grading to capture RO(C3)-graded phenomena and the additional fixed-point structure. The authors develop a stratified strategy combining isotropy separation, the AHSS for BC3 and its BΣ3/X subquotients, and 3-primary Mahowald invariants to assemble π^{C3}_{i,j} from classical 3-primary data, with careful attention to differentials, extensions, and a_Yright-actions. They establish explicit identifications for i<0, relate π^{C3} to π^{cl}(BC3)^ op and to π^{cl} via Φ^{C3} and Res, and provide extensive charts and tables capturing the 3-primary landscape of the C3-stem problem. The results lay groundwork for extending C3-equivariant calculations to broader ranges and offer a framework for integrating Tate-type spectral sequences and Borel deformations in odd-prime equivariant contexts, with potential implications for higher-order Tate spectral sequences and mixed-prime group computations.

Abstract

We compute the spoke-graded $C_3$-equivariant stable homotopy groups of spheres $π_{i, j}^{C_3}$, for stems less than 25 (i.e. $i\leq 25$) and for weights between -16 and 16 (i.e. $-16\leq j\leq 16$). In particular, for $j=2k$, this corresponds to the usual $RO(C_3)$-graded homotopy groups of spheres $π^{C_3}_{i-j+kλ}$ for some fixed 2-dimensional $C_3$-faithful representation $λ$. We also describe the geometric fixed point map $Φ^{C_3}: π_{i, j}^{C_3}\to π_{i-j}^{cl}$ and the underlying map $Res: π_{i, j}^{C_3}\to π_{i}^{cl}$.

$C_3$-equivariant stable stems

TL;DR

This work delivers the first detailed computation of the 3-primary spoke-graded C3-equivariant stable stems π^{C3}_{i,j} for i ≤ 25 and −16 ≤ j ≤ 16, introducing spoke-grading to capture RO(C3)-graded phenomena and the additional fixed-point structure. The authors develop a stratified strategy combining isotropy separation, the AHSS for BC3 and its BΣ3/X subquotients, and 3-primary Mahowald invariants to assemble π^{C3}_{i,j} from classical 3-primary data, with careful attention to differentials, extensions, and a_Yright-actions. They establish explicit identifications for i<0, relate π^{C3} to π^{cl}(BC3)^ op and to π^{cl} via Φ^{C3} and Res, and provide extensive charts and tables capturing the 3-primary landscape of the C3-stem problem. The results lay groundwork for extending C3-equivariant calculations to broader ranges and offer a framework for integrating Tate-type spectral sequences and Borel deformations in odd-prime equivariant contexts, with potential implications for higher-order Tate spectral sequences and mixed-prime group computations.

Abstract

We compute the spoke-graded -equivariant stable homotopy groups of spheres , for stems less than 25 (i.e. ) and for weights between -16 and 16 (i.e. ). In particular, for , this corresponds to the usual -graded homotopy groups of spheres for some fixed 2-dimensional -faithful representation . We also describe the geometric fixed point map and the underlying map .
Paper Structure (30 sections, 51 theorems, 171 equations, 10 figures, 3 tables)

This paper contains 30 sections, 51 theorems, 171 equations, 10 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Strategy
  3. Future work
  4. Organizations
  5. Some non-equivariant background
  6. The classical 3-primary stable stems
  7. Some backgrounds on the Adams $E_2$-page
  8. Some Toda brackets
  9. The 3-primary stunted lens spaces
  10. The cell structures of the stunted lens spaces
  11. The spoke-grading in $C_3$-equivariant homotopy
  12. The Atiayh-Hirzebruch spectral sequence
  13. The $\pi_*^{cl}$-Atiyah--Hirzebruch spectral sequence
  14. Atiyah--Hirzebruch differentials based on the $\mathcal{A}(1)$-module structure
  15. Differentials based on longer attaching maps
  16. ...and 15 more sections

Key Result

Theorem 1.2

For $0\leq i\leq 25$ and $-16\leq j\leq 16$, the 3-primary group structures of $\pi_{i, j}^{C_3}$ are summarized in ChartC_3stablestem. For $i<0$, $\pi^{C_3}_{i,j} \cong \pi^{cl}_{i-j}$, where the latter term denotes the classical stable stems. The torsion-free and $p$-primary ($p\neq 3$) informatio

Figures (10)

  • Figure 1: The $\mathcal{A}(1)$-action on $H^*(BC_{3})^{20}_{1}$
  • Figure 2: The $\mathcal{A}$-module structure and the cell diagram of $C3\wedge C\alpha_1$
  • Figure 3: $\mathcal{A}$-module structure and the cell diagram of $(B\Sigma_3)_3^8$
  • Figure 4: A cell-diagram of $(BC_3)_{-9}^9$ and the splitting
  • Figure 5: The cell structure of $F_2'$.
  • ...and 5 more figures

Theorems & Definitions (113)

  • Definition 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • ...and 103 more