mapping telescope

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A mapping telescope takes a sequence of spaces and maps, $X_1 \xrightarrow{f_1} X_2 \xrightarrow{f_2} \dots$, and builds a new space by gluing cylinders end-to-end according to the maps.

Example. For a fixed dimension $n \ge 1$, let every space $X_i$ be the $n$-sphere, $S^n$. Let the connecting map $f_i: S^n \to S^n$ be a map of degree $i$. The homology groups of the resulting infinite telescope, $M$ is
$$H_p(M) = \begin{cases} \mathbb{Z} & \text{if } p=0 \\ \mathbb{Q} & \text{if } p=n \\ 0 & \text{otherwise} \end{cases}$$

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The key to this computation is a fundamental theorem: the homology of a mapping telescope is the direct limit of the homology groups of the spaces involved. This means we just need to analyze the sequence of homomorphisms induced by our maps:
$$H_p(S^n) \xrightarrow{(f_1)_*} H_p(S^n) \xrightarrow{(f_2)_*} H_p(S^n) \xrightarrow{(f_3)_*} \cdots$$
We know that the homology of $S^n$ is $\mathbb{Z}$ in dimensions $0$ and $n$, and zero otherwise.

• For any dimension $p$ that is not $0$ or $n$, the sequence is just $0 \to 0 \to \dots$, so the limit is trivially $0$.
• In dimension $p=0$, $H_0(S^n) = \mathbb{Z}$ represents its path-connectedness. Any map between connected spaces induces the identity map on $H_0$. Our sequence becomes $\mathbb{Z} \xrightarrow{\text{id}} \mathbb{Z} \xrightarrow{\text{id}} \cdots$, whose direct limit is $\mathbb{Z}$.
• The exciting part is in dimension $p=n$. Here, $H_n(S^n) = \mathbb{Z}$. The map $(f_i)_*$ induced by a map of degree $i$ is multiplication by $i$. This gives us the sequence:
  $$\mathbb{Z} \xrightarrow{\times 1} \mathbb{Z} \xrightarrow{\times 2} \mathbb{Z} \xrightarrow{\times 3} \mathbb{Z} \xrightarrow{\times 4} \cdots$$
  What is the direct limit of this system? An element in the $k$-th group can be thought of as an integer, and the map to the next group multiplies it by $k$. In the limit, this has the effect of allowing division by any integer. For example, the element $1$ in the third group is equivalent to $3$ in the fourth group, which is equivalent to $3 \cdot 4 = 12$ in the fifth group, and so on. This process of allowing division by all non-zero integers is precisely how one constructs the rational numbers $\mathbb{Q}$ from the integers $\mathbb{Z}$.

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Let's call the direct limit of this system $G$. To show that $G$ is isomorphic to $\mathbb{Q}$, we can construct an explicit homomorphism. An element of $G$ is an equivalence class of pairs $[(a,k)]$, where $a$ is an integer from the $k$-th group in the sequence. Let's define a map $\Psi: G \to \mathbb{Q}$ by the rule:
$$\Psi([(a,k)]) = \frac{a}{(k-1)!}$$This map is well-defined because if we move an element a from the $k$-th group to the ($k+1$)-th group, it becomes $k \cdot a$. Their images under $\Psi$ are $\frac{a}{(k-1)!}$ and $\frac{k \cdot a}{k!}$, which are equal. This map $\Psi$ can be shown to be a group isomorphism. It's surjective because any rational $p/q$ can be found as the image of an element, and it's injective because only 0 maps to 0.
So, by gluing spheres together with maps of increasing degree, we've constructed a space whose n-th homology group is the rational numbers.