The Brown-Peterson spectrum is not ${\mathbb{E}}_{2(p^2+2)}$ at 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 $\mathrm{BP}⟨n⟩$ of height $n\ge 4$. This extends results of Lawson at the prime $2$.

arXiv commentv2: Errors corrected and exposition improved. 28 pages