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Let $E_\ast$ be a homology theory and $f: X \to Y$ a map of spaces. Suppose that $\Omega f: \Omega X \to \Omega Y$ is an $E_\ast$-equivalence. Then is $\Omega^2 f: \Omega^2 X \to \Omega^2 Y$ an $E_\ast$-equivalence?

This is true for ordinary integral homology, because if any map $\Omega X \to \Omega Y$ is an $H\mathbb Z$-equivalence, it is a homotopy equivalence by the homology Whitehead theorem (since loop spaces are simple). Alternatively (and more generally), a Serre spectral sequence argument shows that it also holds for $HR$-equivalences at least if $R$ is a PID (to use the universal coefficient theorem).

But is it true, say, for Morava K-theory?

(I have an abstract argument that it is true in general, but I don't trust it very much because it seems like the sort of fact which should hold for very classical reasons if it's true.)

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    $\begingroup$ No luck: K(Z, n+2+m) —> * is a K(n)-equivalence for m ≥ 0, even after passing to k-fold loopspace for k ≤ m, but not when passing to the (k+1)th loopspace. You’d expect your property to hold whenever E has a convergent Eilenberg-Moore spectral sequence. K(n) actually mostly does have an EMSS (arxiv.org/abs/0803.3798), but there’s an exception to its convergence, which this example exploits. $\endgroup$ – Eric Peterson Apr 28 at 2:55
  • $\begingroup$ @EricPeterson Thanks, I totally should have pieced this one together. $\endgroup$ – Tim Campion Apr 28 at 2:58
  • $\begingroup$ No trouble :) The Bauer reference (+ your other questions about virtual homology) tells you pretty precisely what you can get away with, which you might still find interesting. $\endgroup$ – Eric Peterson Apr 28 at 3:00
  • $\begingroup$ It seems to me the conditions for convergence of the $K(n)$- EMSS are actually quite restrictive: a space has to be truncated. So the EMSS can't be used at all for highly connected spaces. $\endgroup$ – Tim Campion Apr 28 at 3:07
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    $\begingroup$ Great, thanks! It's illuminating to me even to see how you framed the question! $\endgroup$ – Tim Campion Apr 28 at 19:08

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