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.



Relating two different approaches to the Atiyah-Hirzebruch Spectral Sequence
For cohomology, this is theorem 3.3 in 


*Maunder, C.R.F.*, [The spectral sequence of an extraordinary cohomology theory](https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/spectral-sequence-of-an-extraordinary-cohomology-theory/A275BD09CBDEF45CC755FC15AA9CA76D), Proc. Camb. Philos. Soc. 59, 567-574 (1963). [ZBL0116.14603](https://zbmath.org/?q=an:0116.14603).



> 
> **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.


