On realizations of the subalgebra a^ℝ(1) of the r-motivic steenrod algebra

In this paper, we show that the finite subalgebra A^ℝ(1), generated by Sq1 and Sq2, of the R-motivic Steenrod algebra A^ℝ can be given 128 different A^ℝ-module structures. We also show that all of these A-modules can be realized as the cohomology of a 2-local finite R-motivic spectrum. The realization results are obtained using an ℝ-motivic analogue of the Toda realization theorem. We notice that each realization of A^ℝ (1) can be expressed as a cofiber of an ℝ-motivic v₁-self-map. The C₂-equivariant analogue of the above results then follows because of the Betti realization functor. We identify a relationship between the RO(C₂)-graded Steenrod operations on a C₂-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 C2-equivariant realizations of A^C2 (1). We find another application of the ℝ-motivic Toda realization theorem: We produce an ℝ-motivic, and consequently a C₂-equivariant, analogue of the Bhattacharya-Egger spectrum Ƶ, which could be of independent interest.