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I have often heard that it is not known whether or not the Brown-Peterson spectrum $BP$ is an $E_\infty$-ring spectrum. Though I see that this is a somewhat natural question to ask, I have often wondered what would the practical consequences of an answer to this question would be.

In particular, I am wondering what would a positive answer to this question mean for homotopy theory. Are there computations we would be able to do that we couldn't do before? What would be the take away if $BP$ did not admit an $E_\infty$-ring structure?

I would also appreciate any references on this problem. Thanks!

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    $\begingroup$ See the first page of Johnson-Noel's "For complex orientations preserving power operations, p-typicality is atypical", which has a brief paragraph about this. $\endgroup$
    – user97187
    Commented Aug 14, 2016 at 21:05
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    $\begingroup$ Since this question was asked, Tyler Lawson has shown that at $p = 2$, $\mathit{BP}$ is not an $E_n$ algebra for $n\ge 12$, thus providing a negative answer to this conjecture at 2. $\endgroup$
    – Arun Debray
    Commented May 19, 2017 at 20:04
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    $\begingroup$ Since Arun's comment, Andrew Senger has shown $BP$ is not $\mathbb{E}_{2(p^2+2)}$ for odd primes, providing a negative answer in general. $\endgroup$
    – Noah Wisdom
    Commented Dec 6, 2023 at 23:40

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I suppose I should try to answer since the question of whether or not $BP$ is an $E_{\infty}$ ring spectrum was Problem 1 of "Problems in infinite loop space theory'', http://www.math.uchicago.edu/~may/PAPERS/16.pdf, written in 1974, two years after $E_{\infty}$ ring spectra were first defined. At the time, we were hoping for some good way of starting from the relevant group law and manufacturing $BP$ from it. The example "in nature'' that led to the definition of $E_{\infty}$ ring spectra was $MU$, so it was impossible not to be intrigued by the question.
Some things proven since with $MU$ might be streamlined if we could start with $BP$, such as the construction of Morava $K$-theory. However, Basterra and Mandell proved that $BP$ is an $E_4$-spectrum, compatibly with $MU$, which is enough for such applications. I'm still intrigued by the question, although I admit to having no applications in mind.

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  • $\begingroup$ Just a personal curiosity, but do you know of any application that requires $BP$ to be more than $E_1$? $\endgroup$
    – Denis Nardin
    Commented Aug 15, 2016 at 12:00
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    $\begingroup$ Well certainly one wants the category of $BP$-modules to be symmetric monoidal under the smash product, which won't come from $E_1$, and one wants to know how which Dyer-Lashof operations are preserved from $MU$ (not all, of course). $\endgroup$
    – Peter May
    Commented Aug 15, 2016 at 23:16
  • $\begingroup$ @PeterMay, what is the general belief amongst homotopy theorists regarding an answer to this question? Is there evidence pointing one way or the other? $\endgroup$
    – CWcx
    Commented Aug 16, 2016 at 20:35
  • $\begingroup$ I doubt there is a consensus. It is $E_4$ compatibly with $MU$, but if it is $E_{\infty}$, then it only is so incompatibly with $MU$. I have no strong feeling, but I would hazard a guess that it is not $E_{\infty}$. $\endgroup$
    – Peter May
    Commented Aug 16, 2016 at 21:38
  • $\begingroup$ It is worth pointing out that if $BP$ is $E_{\infty}$ then Andy Baker has constructed it (if I recall correctly). arxiv.org/abs/1204.4878 Also, @PeterMay, is this arxiv.org/pdf/1310.3336v1.pdf the reference you had in mind with your above comment about the Quillen idempotent being $E_4$? I don't recall them getting more than $E_3$, but maybe you mean a different paper. $\endgroup$
    – Sean Tilson
    Commented Aug 24, 2016 at 11:35

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