Question

Let $G$ and $H$ be torsion abelian groups. Are the following are equivalent:

1. $\mathrm{Hom}(G, H) = 0$

2. $\mathrm{map}_*( K(G, m), K(H,n)) \sim *$ for all $n, m\geq 1$

?

Clearly (2) implies (1).

Answer 1

No, 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).