We provide a lower bound for the coherence of the homotopy commutativity of the Brown–Peterson spectrum, , at a given prime and prove that it is at least –homotopy commutative. We give a proof based on Dyer–Lashof operations that cannot be a Thom spectrum associated to –fold loop maps to for at and 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-–theory.
Birgit Richter. "A lower bound for coherences on the Brown–Peterson spectrum." Algebr. Geom. Topol. 6 (1) 287 - 308, 2006. https://doi.org/10.2140/agt.2006.6.287