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A question from Introduction to Topological Manifolds:

4-28. Suppose $M$ is a noncompact manifold of dimension $n \ge 1$. Show that its one-point compactification is an $n$-manifold if and only if there exists a precompact open subset $U \subseteq M$ such that $M \setminus U$ is homeomorphic to $\mathbb{R}^n \setminus \mathbb{B}^n$. [Hint: you may find the inversion map $f: \mathbb{R}^n \setminus \mathbb{B}^n \to \overline{\mathbb{B}^n}$ defined by $f(x)=x/|x|^2$ useful.]

Here $\mathbb{B}^n$ is the open unit ball in $\mathbb{R}^n$. Can anyone provide a bigger hint for this question?

Denote the one-point compactification of $M$ by $M^*$.

  1. If $M^*$ is an $n$-manifold, then we can choose some neighborhood $E$ around $\infty$ such that $E$ is homeomorphic to $\mathbb{B}^n$. We know that $M^* \setminus E$ is compact. Choose $U = \operatorname{Int} (M^* \setminus E)$; then $$M \setminus U = \overline{M \setminus (M^* \setminus E)} = \overline{E \setminus \{\infty\}}.$$ Where do I go from here?
  2. For the converse, the set $M^* \setminus \overline{U} = \operatorname{Int} (M^* \setminus U)$ is a neighborhood of $\infty$. But how do I connect this with $M \setminus U$?
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For 1, yes you have an $E$ around $\infty$ with $E$ homeomorphic to a ball, but you don't know anything about $\overline{E}$. In fact, $\overline{E}$ is potentially all of $M^\ast$!

On the other hand, inside of $E$ is another open set $E'$, also homeomorphic to a ball, but whose closure is homeomorphic to $\overline{\mathbb{B}^n}$. Use $U = M^\ast \setminus \overline{E'}$.

For 2, since $M\setminus U$ is homeomorphic to $\mathbb{R}^n\setminus \mathbb{B}^n$, it follows that $M\setminus\overline{U}$ is homeomorphic to $\mathbb{R}^n\setminus \overline{\mathbb{B}^n}$. Call such a homeomorphism $g$.

By using the inversion map $f$, one sees that $\mathbb{R}^n\setminus\overline{\mathbb{B}^n}$ is homeomorphic to $\mathbb{B}^n \setminus\{\vec{0}\}$.

Composing $g$ and $f$, we have a homeomorphism between $M\setminus\overline{U}$ and $\mathbb{B}^n\setminus \{\vec{0}\}$. Try to prove that we can use these to find a homeomorphism between $M^\ast \setminus \overline{U}$ and $\mathbb{B}^n$.

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