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

Abstract

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