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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?

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  • $\begingroup$ "the set $[X,Y]$ of homotopy classes of maps from $X$ to $Y$ can be endowed with an abelian group structure." How do you propose to do this? I certainly do not see a natural way to do this (even if we constrict to pointed spaces) . What abelian group is $[S^1,S^1\vee S^1]$? What group is $[S^1 \vee S^1,S^1 \vee S^1]$? $\endgroup$ – PVAL-inactive Aug 18 '14 at 2:03
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    $\begingroup$ This can't work for spaces since $[X,Y]$ is in general only a set. If you work with spectra instead of spaces (the very open) Freyd's generating hypothesis is that for finite spectra $X$ and $Y$ the natural map $[X,Y] \to \text{Hom}_{\pi_* S}(\pi_*X,\pi_* Y)$ is a monomorphism (which Freyd shows actually implies it is an isomorphism). $\endgroup$ – Drew Aug 18 '14 at 2:05
  • $\begingroup$ How do you define the composite of two elements in $[X,Y]$? $\endgroup$ – Hamou Aug 18 '14 at 2:05
  • $\begingroup$ @Drew is right. I am actually interested in spectra, in which case a group structure does exist. I'll edit the question. $\endgroup$ – Alex Turzillo Aug 18 '14 at 2:08
  • $\begingroup$ The generating hypothesis is exactly what I was looking for. Thanks! $\endgroup$ – Alex Turzillo Aug 18 '14 at 2:13
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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.

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