The cohomology of $C_2$-equivariant $A(1)$ and the homotopy of $ko_{C_2}$

TL;DR

This paper computes the cohomology of a key subalgebra of the $C_2$-equivariant Steenrod algebra, providing essential input for understanding the homotopy groups of the equivariant spectrum $ko_{C_2}$ using spectral sequences.

Contribution

It introduces a novel computation of the cohomology of $A^{C_2}(1)$, leveraging motivic calculations to advance equivariant homotopy theory.

Findings

Computed the cohomology of $A^{C_2}(1)$

Connected motivic and equivariant calculations

Facilitated the analysis of $RO(C_2)$-graded homotopy groups

Abstract

We compute the cohomology of the subalgebra $A^{C_2}(1)$ of the $C_2$-equivariant Steenrod algebra $A^{C_2}$. This serves as the input to the $C_2$-equivariant Adams spectral sequence converging to the $RO(C_2)$-graded homotopy groups of an equivariant spectrum $ko_{C_2}$. Our approach is to use simpler $\mathbb{C}$-motivic and $\mathbb{R}$-motivic calculations as stepping stones.