3
$\begingroup$

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?

$\endgroup$
2
  • $\begingroup$ 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, 2016 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...) $\endgroup$ Jun 13, 2016 at 19:13

1 Answer 1

3
$\begingroup$

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).

$\endgroup$
1
  • $\begingroup$ 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, 2016 at 6:52

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.