C_2-Equivariant Stable Stems

TL;DR

This paper computes specific low-stem $C_2$-equivariant stable homotopy groups, providing new data that, combined with existing computations, advances understanding of equivariant homotopy theory up to stem 20.

Contribution

It provides explicit calculations of $C_2$-equivariant stable homotopy groups for stems up to 25 and relates them to classical groups, extending known results in equivariant topology.

Findings

Computed $ ext{pi}^{C_2}_{s,c}$ for $0 \\leq s \\leq 25$ and $-1 \\leq c \\leq 7$

Determined the forgetful map to classical stable homotopy groups in the same range

Predicted all $C_2$-equivariant groups up to stem 20 using periodicity and motivic computations

Abstract

We compute the 2-primary $C_2$-equivariant stable homotopy groups $\pi^{C_2}_{s,c}$ for stems between 0 and 25 (i.e., $0 \leq s \leq 25$) and for coweights between -1 and 7 (i.e., $-1 \leq c \leq 7)$. Our results, combined with periodicity isomorphisms and sufficiently extensive $\mathbb{R}$-motivic computations, would determine all of the $C_2$-equivariant stable homotopy groups for all stems up to 20. We also compute the forgetful map $\pi^{C_2}_{s,c} \rightarrow \pi^{\mathrm{cl}}_s$ to the classical stable homotopy groups in the same range.