On a Spectrum-level Splitting of the $BP \langle 2 \rangle$-Cooperations Algebra

TL;DR

This paper constructs a spectrum-level splitting of the $BP angle 2 angle$-cooperations algebra, extending previous work on lower spectra to facilitate computations in stable homotopy theory.

Contribution

It introduces a new splitting for $BP angle 2 angle wedge BP angle 2 angle$, enabling advanced calculations in the Adams spectral sequence.

Findings

Constructed a spectrum-level splitting for $BP angle 2 angle wedge BP angle 2 angle$

Facilitates computations in the $BP angle 2 angle$-based Adams spectral sequence

Extends classical splittings from $bo$ and $BP angle 1 angle$ to $BP angle 2 angle$

Abstract

In the 1980s, Mahowald and Kane used Brown-Gitler spectra to construct splittings of $bo \wedge bo$ and $BP\langle 1 \rangle \wedge BP\langle 1 \rangle$. These splittings helped make it feasible to do computations using the $bo$- and $BP \langle 1 \rangle$-based Adams spectral sequences. In this paper, we construct an analogous splitting for $BP \langle 2 \rangle \wedge BP\langle 2 \rangle$.