A <direct system of abelian groups> over the directed poset $A$ consists of groups $G_a$ and maps $\phi_{ab}:G_a\to G_b$ for $a\leq b$, satisfying $\phi_{aa}=1$ and $\phi_{bc}\phi_{ab}=\phi_{ac}$. Its <direct limit> is the quotient of $\bigoplus_aG_a$ by the relations $g_a\sim\phi_{ab}(g_a)$.

For the displayed sequence, put $P_0=1$ and $P_n=a_0a_1\cdots a_{n-1}$. Map the copy of $\mathbb Z$ at stage $n$ to $\mathbb Q$ by
$$
m\longmapsto \frac{m}{P_n}.
$$
This is compatible with the next transition because $a_nm/P_{n+1}=m/P_n$. The universal property of the <direct limit> therefore identifies it with
$$
\boxed{\bigcup_{n\geq0}\frac1{P_n}\mathbb Z.}
$$
In reduced form, these are exactly the rationals whose denominator divides one of the finite products $P_n$. This is the <sequential direct limit of multiplication maps on the integers>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 114 / 2 / 1 / Solution

List the prime numbers without repetition as $p_0,p_1,p_2,\ldots$. Choose maps
$$
S^n\xrightarrow{f_i}S^n,
\qquad \deg f_i=p_i,
$$
and let $T$ be their <mapping telescope>. A finite initial telescope deformation retracts onto its last sphere. The inclusions of successive finite telescopes induce multiplication by $p_i$ on degree-$n$ reduced homology. The <homology of a directed union> and the <sequential direct limit of multiplication maps on the integers> therefore give
$$
\widetilde H_j(T;\mathbb Z)
\cong
\begin{cases}
\varinjlim(\mathbb Z\xrightarrow{p_0}\mathbb Z\xrightarrow{p_1}\cdots),&j=n,\\
0,&j\ne n.
\end{cases}
$$
Every finite product $p_0\cdots p_r$ is square-free, and every square-free denominator divides one such product. Hence
$$
\boxed{\widetilde H_j(T;\mathbb Z)\cong
\begin{cases}
\mathbb Q_{\mathrm{sq}},&j=n,\\
0,&j\ne n,
\end{cases}}
$$
as in the <mapping-telescope realization of the rational group with square-free denominators>.

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 114 / 2 / 2 / 3 / Solution

{title2=$\varinjlim(\mathbb Z\xrightarrow{a_0}\mathbb Z\xrightarrow{a_1}\cdots)$}

For nonzero integers $a_i$ and $P_n=a_0\cdots a_{n-1}$, the compatible embeddings of stage $n$ into $\mathbb Q$ send $m$ to $m/P_n$. They identify the direct limit with
$$
\bigcup_{n\geq0}P_n^{-1}\mathbb Z\subseteq\mathbb Q.
$$


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 Incoming links: Sequential direct limit of multiplication maps on the integers