On realizations of the subalgebra $A^R(1)$ of the $R$-motivic Steenrod Algebra

In this paper, we show that the finite subalgebra $\mathcal{A}^{\mathbb{R}}(1)$, generated by $\mathrm{Sq}^1$ and $\mathrm{Sq}^2$, of the $\mathbb{R}$-motivic Steenrod algebra $\mathcal{A}^{\mathbb{R}}$ can be given $128$ different $\mathcal{A}^{\mathbb{R}}$-module structures. We also show that all of these $\mathcal{A}^{\mathbb{R}}$-modules can be realized as the cohomology of a $2$-local finite $\mathbb{R}$-motivic spectrum. The realization results are obtained using an $\mathbb{R}$ -motivic analogue of the Toda realization theorem. We notice that each realization of $\mathcal{A}^{\mathbb{R}}(1)$ can be expressed as a cofiber of an $\mathbb{R}$-motivic $v_1$-self-map. The $\mathrm{C}_2$-equivariant analogue of the above results then follows because of the Betti realization functor. We identify a relationship between the $\mathrm{RO}(\mathrm{C}_2)$-graded Steenrod operations on a $\mathrm{C}_2$-equivariant space and the classical Steenrod operations on both its underlying space and its fixed-points. This technique is then used to identify the geometric fixed-point spectra of the $\mathrm{C}_2$-equivariant realizations of $\mathcal{A}^{\mathrm{C}_2}(1)$. We find another application of the $\mathbb{R}$-motivic Toda realization theorem: we produce an $\mathbb{R}$-motivic, and consequently a $\mathrm{C}_2$-equivariant, analogue of the Bhattacharya-Egger spectrum $\mathcal{Z}$, which could be of independent interest.