Given a (for simplicity connective) spectrum $E$ and a pointed CW-space $X$ there is "the" (homological) Atiyah-Hirzebruch spectral sequence

$$E_{pq}^2 = \tilde{H}_p( X, \pi_q(E)) \Rightarrow \pi_{p+q}(X \wedge E)$$

I know of two approaches at arriving at this spectral sequence:

  • One is using a CW-structure $\emptyset \subset X_0 \subset \cdots \subset X_s \subset X_{s+1} \subset \cdots $ for $X$. Each pair $X_{s-1} \subset X_s$ gives the long exact sequence $$ \cdots \rightarrow \tilde H_n (X_{s-1}) \rightarrow \tilde H_n (X_{s}) \rightarrow H_n (X_{s}, X_{s-1}) \rightarrow \tilde H_n (X_{s-1}) \rightarrow \cdots $$ which we can together assemble into the exact couple $ \bigoplus H_n (X_{s}, X_{s-1}) \rightarrow \bigoplus \tilde H_n (X_{s}) \rightarrow \bigoplus \tilde H_n (X_{s}) \rightarrow \bigoplus H_n (X_{s}, X_{s-1})$. Taking the derived couple and setting $p+q = n, s = q$ gives the above second page of the spectral sequence and with a bit of work one sees that it does converge to the right answer. However, while the second page and the limit do not depend on the chosen CW-Structure for $X$, it is my understanding that the differentials and the induced filtration of $\pi_{p+q}(X \wedge E)$ do.
  • The second is given by taking the Postnikov tower of $E$ $$ \cdots P_s E \rightarrow P_{s-1} E \rightarrow \cdots \rightarrow P_1 E \rightarrow P_0E \rightarrow * $$ The maps $P_s E \rightarrow P_{s-1}E$ have as homotopy fibers shifted Eilenberg Maclane spectra $\Sigma^s H \pi_s(E)$. Smashing with $X$ gives the distinguished triangles $$ X \wedge \Sigma^s H \pi_s(E) \rightarrow X \wedge P_s E \rightarrow X \wedge P_{s-1} E \rightarrow X \wedge \Sigma^{s+1} H \pi_s(E)$$ Note that $\pi_n(X \wedge \Sigma^s H \pi_s(E)) = \tilde H_{n-s}(X,\pi_s E) $. Taking the induced long exact sequences in homotopy groups and assembling them together similarly to before we get the exact couple $\bigoplus \tilde H_{n-s}(X,\pi_s E) \rightarrow \bigoplus \pi_n (X \wedge P_s E) \rightarrow \bigoplus \pi_n (X \wedge P_s E) \rightarrow \bigoplus \tilde H_{n-s}(X,\pi_s E) $ Again setting $n = p+q, s=q$, we already have the second page of the spectral sequence, and using that $E$ is the homotopy limit of the $P_n E$'s gives that the spectral sequence converges to the right terms.

While it is clear that both spectral sequences compute the same thing, it is not clear to me in what sense they are the same spectral sequence - The Postnikov-tower is unique up to weak equivalence, so the resulting spectral sequence does not depend on this choice, however the choices made in the first case (of a CW-structure for X) seem to make a qualitative difference in how to compute the spectral sequence and I wouldn't expect the filtrations I get in each case to have anything to do with each other. Has this problem already been looked at? I'd be thankful for a guide to the right literature.

  • 1
    Maybe it is worth pointing out that what you are seeing in the dependence on the CW structure is like how the $E_1$-page depends on a choice of resolution but $E_2$-page doesn't have such a dependence. – Sean Tilson Mar 13 at 13:05
up vote 12 down vote accepted

For cohomology, this is theorem 3.3 in

Maunder, C.R.F., The spectral sequence of an extraordinary cohomology theory, Proc. Camb. Philos. Soc. 59, 567-574 (1963). ZBL0116.14603.

Theorem 3.3 If $E_r^{p,q}$ is the spectral sequence associated to a skeletal filtration and $\bar E_r^{p,q}$ is the spectral sequence associated to the Postnikov tower, there exist for all $r\ge 2$ isomorphisms $\phi_r:E_r^{p,q}\to \bar E_r^{p,q}$ such that $$ \phi_r d_r = d_r\phi_r\,.$$

In fact (see ibid., lemma 4.3.1) the proof of the theorem gives an isomorphism of the corresponding exact couples (after they are both indexed to give the $E_2$-page).

I believe the same proof gives the corresponding statement for the homology AHSS.

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