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 $v_1$selfmap. 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 fixedpoints. This technique is then used to identify the geometric fixedpoint 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 BhattacharyaEgger spectrum $\mathcal{Z}$, which could be of independent interest.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.10769
 Bibcode:
 2021arXiv210610769B
 Keywords:

 Mathematics  Algebraic Topology;
 14F42;
 55S10;
 55N91
 EPrint:
 Minor changes. Submitted version