On realizations of the subalgebra $\mathcal {A}^{\mathbb {R}}(1)$ of the $\mathbb {R}$-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 $\mathcal {A}$-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.