# What's wrong with this argument that the Atiyah-Hirzebruch spectral sequence always degenerates?

Let $$E$$ be a spectrum and let $$X$$ be a space or connective spectrum. Then the cohomological Atiyah-Hirzebruch spectral sequence is of the form:

$$H^s(X,E^t) \Rightarrow E^{s+t}(X)$$

This is a half-plane spectral sequence with entering differentials. If $$E' \to E$$ is a map of spectra, then I believe I get a map of spectral sequences to this from the analogous spectral sequence for $$(E')^\ast(X)$$. In particular, let $$E' = \tau_{\geq k} E$$ be a connective cover. Then for degree reasons, any differential $$d_u^{k,t}$$ vanishes in the $$E'$$-AHSS. By mapping forward along the map of spectral sequences, this seems to imply that $$d_u^{k,t}$$ also vanishes in the $$E$$-AHSS. But since this holds for all $$k \in \mathbb Z$$, we seem to have concluded that the $$E$$-AHSS collapses, which can't be right at all.

Where did this argument go wrong?

• I think I see -- I messed up the grading. We have $E^\ast = E_{-\ast}$, so $(\tau_{\geq k} E)^\ast$ is concentrated in degrees $\leq -k$, not $\geq k$. Thus the differentials are entering from the zero region, and there's actually no information transferred in the map of spectral sequences. – tcamps May 25 at 12:53