The Brown-Peterson spectrum is not $\mathbb{E}_{2(p^2+2)}$ at odd primes

TL;DR

This paper proves that the odd-primary Brown-Peterson spectrum cannot have certain high-level structured ring spectrum properties, extending known results at the prime 2 to odd primes.

Contribution

It establishes new non-existence results for $ ext{E}_n$ structures on Brown-Peterson spectra at odd primes, generalizing previous prime 2 findings.

Findings

Brown-Peterson spectrum lacks $ ext{E}_{2(p^2+2)}$ structure at odd primes

No $ ext{E}_{2p+3}$ map from MU to BP at odd primes

Results extend Lawson's prime 2 results to odd primes

Abstract

We show that the odd-primary Brown-Peterson spectrum $\mathrm{BP}$ does not admit the structure of an $\mathbb{E}_{2(p^2+2)}$ ring spectrum and that there can be no map $\mathrm{MU} \to \mathrm{BP}$ of $\mathbb{E}_{2p+3}$ ring spectra. We also prove the same results for truncated Brown-Peterson spectra $\mathrm{BP} \langle n \rangle$ of height $n \geq 4$. This extends results of Lawson at the prime $2$.