An $R$-motivic $v_1$-self-map of periodicity $1$

We consider a nontrivial action of $\mathrm{C}_2$ on the type $1$ spectrum $\mathcal{Y}:=\mathcal{M}_2(1) \wedge \mathrm{C}(\eta)$, which is well-known for admitting a $1$-periodic $v_1$-selfmap. The resultant finite $\mathrm{C}_2$-equivariant spectrum $\mathcal{Y}^{\mathrm{C}_2}$ can also be viewed as the complex points of a finite $\mathbb{R}$-motivic spectrum $\mathcal{Y}^\mathbb{R}$. In this paper, we show that one of the $1$-periodic $v_1$-self-maps of $\mathcal{Y}$ can be lifted to a self-map of $\mathcal{Y}^{\mathrm{C}_2}$ as well as $\mathcal{Y}^\mathbb{R}$. Further, the cofiber of the self-map of $\mathcal{Y}^\mathbb{R}$ is a realization of the subalgebra $\mathcal{A}^\mathbb{R} (1)$ of the $\mathbb{R}$-motivic Steenrod algebra. We also show that the $\mathrm{C}_2$-equivariant self-map is nilpotent on the geometric fixed-points of $\mathcal{Y}^{\mathrm{C}_2}$.

Keywords

self-map, motivic homotopy, equivariant homotopy

2010 Mathematics Subject Classification

14F42, 55Q51, 55Q91

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B. Guillou and A. Li were supported by NSF grants DMS-1710379 and DMS-2003204.

Received 30 October 2020

Received revised 22 February 2021

Accepted 15 March 2021

Published 13 April 2022