$C_3$-equivariant stable stems

TL;DR

This paper computes specific $C_3$-equivariant stable homotopy groups of spheres for stems less than 25 and weights between -16 and 16, providing detailed algebraic and geometric insights.

Contribution

It provides explicit calculations of $C_3$-equivariant stable homotopy groups and describes the geometric fixed point and underlying maps for these groups.

Findings

Computed $ ext{pi}_{i,j}^{C_3}$ for stems $i extless 25$ and weights $-16 extless j extless 16$.

Described the geometric fixed point map $ ext{ extPhi}^{C_3}$ and the restriction map in this context.

Clarified the relation between equivariant and classical homotopy groups for these cases.

Abstract

We compute the spoke-graded $C_3$-equivariant stable homotopy groups of spheres $\pi_{i, j}^{C_3}$, 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 $\pi^{C_3}_{i-j+k\lambda}$ for some fixed 2-dimensional $C_3$-faithful representation $\lambda$. We also describe the geometric fixed point map $\Phi^{C_3}: \pi_{i, j}^{C_3}\to \pi_{i-j}^{cl}$ and the underlying map $Res: \pi_{i, j}^{C_3}\to \pi_{i}^{cl}$.