A lower bound for coherences on the Brown–Peterson spectrum

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 (2p2 + 2p− 2)–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.