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

Abstract

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.

Minor changes. Submitted version

References in corpus (2)

• The telescope conjecture at height 2 and the tmf resolution
• An $\mathbb{R}$-motivic $v_1-$self-map of periodicity $1$