Question about spectral sequences

Question

I have a somewhat naive question about spectral sequences. Suppose I have a generalized homology theory $E_*$ and want to compute $E_*(X)$ for some space X. A common tool for this type of question is spectral sequences. Say I use the Adams spectral sequence and that it collapses to the second page. Given this, if I were instead to use the Atiyah-Hirzebruch spectral sequence to compute $E_*(X)$ would I know that it also collapses to the second page?

Answer 1

Here is a simple example demonstrating that degeneration of the classical mod $p$ Adams spectral sequence does not imply degeneration of the corresponding Atiyah-Hirzebruch spectral sequence.

Let's try to compute $\pi_*(H\mathbb{F}_p) \cong \mathbb{F}_p$. The $H\mathbb{F}_p$-based Adams spectral sequence has signature $$E_2^{s,t} = \operatorname{Ext}_{\mathcal{A}_p}^{s,t}(H\mathbb{F}_p^* H\mathbb{F}_p, \mathbb{F}_p) \Rightarrow \pi_{t-s} H\mathbb{F}_p.$$ Since $H\mathbb{F}_p^* H\mathbb{F}_p \cong \mathcal{A}_p$ is free over itself, the $E_2$-page is just $\mathbb{F}_p$ concentrated in bidegree $(0,0)$, and the spectral sequence collapses.

On the other hand, the Atiyah-Hirzebruch spectral sequence computing $\pi_* H\mathbb{F}_p$ has signature $$E^2_{s,t} = H_s(H\mathbb{F}_p; \pi_t) \Rightarrow \pi_{s+t} H\mathbb{F}_p.$$ In this case, the $E_2$-page is extremely large, involving both various Steenrod algebras and the stable homotopy groups of spheres. In particular, this spectral sequence does not degenerate at $E^2$.