A lower bound for coherences on the Brown–Peterson spectrum

Abstract

We provide a lower bound for the coherence of the homotopy commutativity of the Brown–Peterson spectrum, $BP$, at a given prime $p$ and prove that it is at least $\left(2p^2+2p-2\right)$–homotopy commutative. We give a proof based on Dyer–Lashof operations that $BP$ cannot be a Thom spectrum associated to $n$–fold loop maps to $BSF$ for $n=4$ at $2$ and $n=2p+4$ at odd primes. Other examples where we obtain estimates for coherence are the Johnson–Wilson spectra, localized away from the maximal ideal and unlocalized. We close with a negative result on Morava-$K$–theory.