Homology, Homotopy and Applications

Volume 24 (2022)

Number 1

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

Pages: 299 – 324

DOI: https://dx.doi.org/10.4310/HHA.2022.v24.n1.a15


Prasit Bhattacharya (Department of Mathematics, University of Notre Dame, Indiana, U.S.A.)

Bertrand Guillou (Department of Mathematics, University of Kentucky, Lexington, Ky., U.S.A.)

Ang Li (Department of Mathematics, University of Kentucky, Lexington, Ky., U.S.A.)


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 $\mathcal{Y}^{\mathrm{C}_2}$.


self-map, motivic homotopy, equivariant homotopy

2010 Mathematics Subject Classification

14F42, 55Q51, 55Q91

Copyright © 2022, Prasit Bhattacharya, Bertrand Guillou and Ang Li. Permission to copy for private use granted.

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