The wedge sum $\bigvee_{k \in 2 \mathbb{Z}} S^{k}$ is an $A_\infty$-ring spectrum: the connective cover is the free $A_\infty$-ring on the sphere $S^2$, if I'm not mistaken, and then one inverts the element in $\pi_2$. It is also homotopy commutative. Can it be made into an $E_\infty$-ring spectrum? More generally, given an $E_\infty$-ring spectrum $R$, when can $\bigvee_{k \in 2 \mathbb{Z}} \Sigma^k E$ be made into an $E_\infty$-ring?
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asked Nov 2, 2012 at 2:23
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6$\begingroup$ I'm pretty sure it is not, why would it be? Sorry not to have time to say more. Essentially, even with the $2$, the commutativity is only homotopical, not as strict as would be required. $\endgroup$– Peter MayCommented Nov 2, 2012 at 3:11
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2$\begingroup$ Here's a point of terminology that will help you find references: an $H_\infty$ structure on $\bigvee_{k\in d\mathbb{Z}}\Sigma^kE$ is essentially what is called an $H^d_\infty$ structure in the $H_\infty$ book (Springer LNM 1176). Of course $H_\infty$ is weaker than $E_\infty$ but this is a start. $\endgroup$– Neil StricklandCommented Nov 2, 2012 at 7:07
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1 Answer
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Ah, tracked it down. Here is an argument. I should mention that Peter once pointed me towards an original source due to McClure, or page 238, Prop. 6.1, of SLN 1176 (the $H_\infty$ book).
Suppose you had such a ring object $R$. We examine its mod-2 homology. This has several features:
It is the ring $\mathbb Z/2[t^{\pm 1}]$.
It has trivial action of the Steenrod operations $P_r$ (this is dual to the cohomology action), because it's a wedge of spheres.
From the $H_\infty$ book, it has Dyer-Lashof operations $Q^s$. These satisfy $Q^{|x|} x = x^2$, and the Nishida relations $$ P_r Q^s = \sum \binom{s - r}{r - 2i} Q^{s-r+i}P_i. $$
In particular, these together would say $$ 0 = P_2 Q^4 t = \binom{2}{2} Q^{2}P_0 t + \binom{2}{0} Q^{4}P_2 t = Q^2 t = t^2. $$
If $E$ is a commutative $MU$-algebra, then you can use the map from $MU$ to its periodic version $MUP$ to produce $R \wedge_{MU}MUP$, which is 2-periodified. Unfortunately, we know very few genuine $MU$-algebras. Barry Walker proved that complex K-theory is one of them, based on Matthew Ando's work on studying $H_\infty$ structures on Lubin-Tate cohomology theories. My understanding is that the problem is still open for almost all of the Lubin-Tate cohomology theories.
answered Nov 2, 2012 at 5:04
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2$\begingroup$ Thank you! LNM 1176 is on my reading list, but I haven't gotten very far in it. These types of methods seem like quite a useful tool for answering such questions. $\endgroup$ Commented Nov 2, 2012 at 11:32
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6$\begingroup$ Perhaps worth pointing out: the ring spectrum in question can be made E_2, since it can be described as the Thom spectrum of a 2-fold loop map Omega^2(BU(1)) -> Omega^2(BU) = Z x BU. $\endgroup$ Commented Nov 4, 2012 at 16:03