I'm interested in some nice identifications (together with explanations) about the edge homomorphisms in the AHSS for a spectrum $X$.

$$\times \times \times $$ Let $X$ be a connective spectrum, as stated in Adams' stable homotopy and generalised homology book for example, for a reduced homology theory $h_*$, there is the AHSS $$H_p(X;h_q(S^0))\Rightarrow h_{p+q}(X)$$ which clearly is a first quadrant spectral sequence. I'm interested in the edge homomorphisms which arise from it, the vertical and the horizontal one.

For a space $Y$ the situation is well-understood and described in the standard literature (Davis & Kirk's book for example), but I can't find anything for a spectrum $X$ (Adams only mentions its existence, not giving any additional details).

My opinion is that the situation for the horizontal edge homomorphism might be messy in the spectra case, since it seems to be general knowledge that the differential $$d_2 \colon H_p(X; \Omega_0^{Spin}) \to H_{p-2}(X;\Omega_1^{Spin})$$ which become $$d_2 \colon H_p(X; \mathbb{Z}) \to H_{p-2}(X;\mathbb{Z}_2)$$ is reduction modulo $2$ composed with the dual of the Steenrod square $Sq^2$, therefore not always trivial (as in the case for a space $Y$ where one can analyse the edge homomorphism). What about the vertical one?


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