#### Volume 6, issue 1 (2006)

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A lower bound for coherences on the Brown–Peterson spectrum

### Birgit Richter

Algebraic & Geometric Topology 6 (2006) 287–308
 arXiv: math.AT/0504322
##### 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 $\left(2{p}^{2}+2p-2\right)$–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.

##### Keywords
structured ring spectra, Brown-Peterson spectrum
Primary: 55P43
Secondary: 13D03