AN ℝ-MOTIVIC v1-SELF-MAP OF PERIODICITY 1.

We consider a nontrivial action of C2 on the type 1 spectrum Y := M2(1) Λ C(η), which is well-known for admitting a 1-periodic v1-self-map. The resultant finite C2-equivariant spectrum Y... can also be viewed as the complex points of a finite ℝ-motivic spectrum Yℝ. In this paper, we show that one of the 1-periodic v1-self-maps of Y can be lifted to a self-map of Y... as well as Yℝ. Further, the cofiber of the self-map of Yℝ is a realization of the subalgebra Aℝ (1) of the ℝ-motivic Steenrod algebra. We also show that the C2-equivariant self-map is nilpotent on the geometric fixed-points of Y.... [ABSTRACT FROM AUTHOR]