# Understanding why the Adams Spectral Sequence works

I am trying to learn about the Adams Spectral Sequence and my question is basically summed up in the title.

More precisely, let $$X$$, $$Y$$, and $$E$$ be spectra. We have a homomorphism $$[X,Y] \to Hom_{E^∗E}(E^∗Y,E^∗X)$$, the latter giving the first page of the Adams Spectral Sequence. But why does $$Ext^∗_{E^∗E}(E^∗Y,E^∗X)$$ give a "better approximation" to $$[X,Y]$$ than $$Hom_{E^∗E}(E^∗Y,E^∗X)$$?

And more generally, it seems like magic that we can get homotopy from successively taking homology. Any intuition about this?

• Pretend that you understand homotopy classes of maps in one case: if the codomain is $E$, then $[X,E] = E^*X$. Then if you want to compute $[X,Y]$ for a general $Y$, you should try to approximate $Y$ by copies of $E$ somehow. This leads to the $E$-Adams tower and then to the Adams SS, which is built up from a sort of resolution of $Y$ by copies of $E$. This at least makes it plausible that Ext is relevant, rather than just Hom, or at least that homological algebra could enter into things. – John Palmieri Dec 3 '18 at 6:46