On realizations of the subalgebra \mathcal{A}^{\mathbb{R}}(1) of the \mathbb{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}-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. [ABSTRACT FROM AUTHOR]