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This question is actually about reading Adams' Stable Homotopy and Generalised Cohomology; in Part II chapter 2, there are two numbered lemmata (Lemma 2.5 contravariant, 2.14 covariant) to the effect that

The Atiyah-Hirzebruch spectral sequences $H_p(\mathbb{CP}^\infty,E_q(*))\Rightarrow E_{p+q}(\mathbb{CP}^\infty)$ and $H_p(\mathbb{CP}^\infty\times \mathbb{CP}^\infty,E_q(*))\Rightarrow E_{p+q}(\mathbb{CP}^\infty\times \mathbb{CP}^\infty)$ collapse at page 2.

Now, read in context, Adams has already mentioned that there are three spectra he's interested in: $H,KU,MU$ (where the AH-SS indeed collapses for sparsity reasons); but there is no mention of this in the statements of either lemma, nor in the (very terse) arguments given.

Question Can someone confirm whether $E\in \{ H, KU , MU \}$ is in fact what Adams meant?

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  • $\begingroup$ Jesse, please give numbers of numbered statements and a more precise reference. A first glance shows me nothing that matches your description. $\endgroup$ – Peter May Jun 13 '16 at 16:47
  • $\begingroup$ OK, so what's going on here is I've confused myself about which Adams book I was looking at; this in part due to someone-else having confused me about which Adams book I was looking at, but... adjusting in a couple of minutes... though I suspect you particularly could still guess what Adams meant better than I. (but guessing is not what we're about, either...) $\endgroup$ – Jesse C. McKeown Jun 13 '16 at 19:13
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I couldn't find the exact reference, but I guess $H$, $KU$ and $MU$ are probably in particular what is meant.

In general, the Atiyah–Hirzebruch spectral sequence collapses for complex oriented multiplicative cohomology theories (cf. e.g. Proposition 7 in Lecture 4 of Lurie's notes on chromatic homotopy theory).

Edit: Now that we have a more precise reference, I found that that in the beginning of that chapter it reads:

We will study [ring] spectra $Ε$ which are provided with "orientations", in the following sense (which owes much to a seminar by A. Dold). There is given an element $x \in \tilde{E}^*(\mathbb{C}P^\infty)$ such that $\tilde{E}^*(\mathbb{C}P^1)$ is a free module over $\pi_*(E)$ on the generator $i^*x$, where $i \colon \mathbb{C}P^1 \to \mathbb{C}P^\infty$ is the inclusion map.

So it looks like what Adams actually means is also a general complex oriented cohomology theory (resp. complex oriented spectrum).

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  • $\begingroup$ You may want to emphasize that the condition on the orientability of $E$ is crucial here. For example, if $E=S$, the SS doesn't collapse at $E_2$. $\endgroup$ – user43326 Jun 15 '16 at 6:52

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