The reduced ring of the $RO(C_2)$-graded $C_2$-equivariant stable stems

TL;DR

This paper provides a detailed algebraic description of the ring structure of $C_2$-equivariant stable stems, focusing on generators, relations, and the structure modulo nilpotent elements, including rational spheres.

Contribution

It offers the first explicit presentation of the $RO(C_2)$-graded $C_2$-equivariant stable stems' ring structure, including rational cases, using generators and relations.

Findings

Ring structure of $oldsymbol{ ilde{ ext{pi}}}_ullet^{C_2}$ described

Homotopy groups of rational $C_2$-sphere characterized

Generators and relations explicitly identified

Abstract

We describe in terms of generators and relations the ring structure of the $RO(C_2)$-graded $C_2$-equivariant stable stems $\pi_\star^{C_2}$ modulo the ideal of all nilpotent elements. As a consequence, we also record the ring structure of the homotopy groups of the rational $C_2$-equivariant sphere $\pi_\star^{C_2}(S_\mathbb{Q})$.