$BP$: Close encounters of the $E_\infty$ kind

TL;DR

This paper constructs an $E_$ ring spectrum $R$ closely approximating the $p$-local Brown-Peterson spectrum $BP$, and shows they are equivalent if $BP$ admits an $E_$ structure, using an inductive cellular approach.

Contribution

It provides a new inductive cellular construction of an $E_$ ring spectrum approximating $BP$, linking cellular methods with power operations.

Findings

Constructed an $E_$ ring spectrum $R$ approximating $BP$.

Proved that if $BP$ has an $E_$ structure, then $R$ and $BP$ are weakly equivalent as $E_$ ring spectra.

Utilized power operations to define homotopy classes and attach $E_$ cells.

Abstract

Inspired by Stewart Priddy's cellular model for the $p$-local Brown-Peterson spectrum $BP$, we give a construction of a $p$-local $E_\infty$ ring spectrum $R$ which is a close approximation to $BP$. Indeed we can show that if $BP$ admits an $E_\infty$ structure then these are weakly equivalent as $E_\infty$ ring spectra. Our inductive cellular construction makes use of power operations on homotopy groups to define homotopy classes which are then killed by attaching $E_\infty$ cells.