A spectrum-level splitting of the $k{u}_{\mathbb{R}}$-cooperations algebra

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 paper, 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 $H\underline{\mathbb{Z}}\wedge H\underline{\mathbb{Z}}$. Along the way, we give complete computations of the $k{u}_{\mathbb{R}}$ and $H\mathbb{Z}$ operations and cooperations algebras.

59 pages, corrects typos, comments welcome