homotopy class of maps in terms of homotopy groups of spectra

Question

Given spectra $X$ and $Y$, the set $[X,Y]$ of homotopy classes of maps from $X$ to $Y$ can be endowed with an abelian group structure. Can the group $[X,Y]$ be expressed in terms of the homotopy groups $\pi_i(X)$ and $\pi_i(Y)$? A naive guess is\begin{equation*}[X,Y]\stackrel{?}{=}\bigoplus^\infty_{i=0}\text{Hom}(\pi_i(X),\pi_i(Y))\end{equation*}but this is not quite right. What is the correct expression, as general as possible?

Answer 1

Since you are interested in spectra then the generating hypothesis is what you want. Namely Freyd conjectured that for finite spectra $X$ and $Y$ the natural map $$ [X,Y] \to \text{Hom}_{\pi_* S}(\pi_*X,\pi_*Y) $$ is a monomorphism. I recommend Hovey's article for some equivalent statements and consequences.