Mapping-telescope realization of the rational group with square-free denominators
= Mapping-telescope realization of the rational group with square-free denominators
{title2=$\widetilde H_n(T)\cong\mathbb Q_{\mathrm{sq}}$}
List the primes as $p_0,p_1,\ldots$ and form the <mapping telescope> of degree-$p_i$ maps $S^n\to S^n$. Its reduced homology vanishes outside degree $n$, while
$$
\widetilde H_n(T;\mathbb Z)\cong\varinjlim(\mathbb Z\xrightarrow{p_0}\mathbb Z\xrightarrow{p_1}\cdots)\cong\mathbb Q_{\mathrm{sq}}.
$$