Abstract

In the 1980’s, Mahowald and Kane used integral Brown-Gitler spectra to decompose $ku \wedge ku$ as a sum of finitely generated $ku$-module spectra. This splitting, along with an analogous decomposition of $ko \wedge ko,$ led to a great deal of progress in stable homotopy computations and understanding of $v_1$-periodicity in the stable homotopy groups of spheres. In this talk, we construct a $C_2$-equivariant lift of Mahowald and Kane’s splitting of $ku \wedge ku$. We also describe the resulting $C_2$-equivariant splitting in terms of $C_2$-equivariant Adams covers and record an analogous splitting for $H \underline{\mathbb{Z}} \wedge H \underline{\mathbb{Z}}$. Similarly to the nonequivariant story, we outline how these techniques facilitate further $C_2$-equivariant stable homotopy computations and understanding of $v_1$-periodicity in $C_2$-equivariant stable stems.