Let $G$ and $H$ be torsion abelian groups. Are the following are equivalent:
$\mathrm{Hom}(G, H) = 0$
$\mathrm{map}_*( K(G, m), K(H,n)) \sim *$ for all $n, m\geq 1$
?
Clearly (2) implies (1).
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Sign up to join this communityNo, we have $Hom(\mathbb{Z}/n,\mathbb{Z})=0$ but there is a nontrivial Bockstein $K(\mathbb{Z}/n,1)\to K(\mathbb{Z},2)$.
For finite abelian groups we have (1) iff (2) iff ($|G|$ and $|H|$ are coprime).