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"Show that every closed ball in $\mathbb{R}^n$ is an $n$-dimensional manifold with boundary, as is the complement of every open ball. Assuming the theorem on the invariance of the boundary, show that the manifold boundary of each is equal to its topological boundary as a subset of $\mathbb{R}^n$, namely a sphere. Hint: for the unit ball in Rn, consider the map $\pi \circ \sigma^{-1}: \mathbb{R}^n \rightarrow \mathbb{R}^n$, where $\sigma$ is the stereographic projection and $\pi$ is a projection from $\mathbb{R}^{n + 1}$ to $\mathbb{R}^n$ that omits some coordinate other than the last."

So, I've got (the first part, anyways) of the question done, but my technique was a little different than what Prof. Lee suggested: I considered the ball as infinitely many foliated spheres, mapped each one to a plane using the stereographic projection, and then put the last coordinate as a function of the distance from the north/south pole. However, my solution seemed to neatly avoid any use of $\pi$ as mentioned, and I'm curious if anyone knows how that solution runs.

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    $\begingroup$ The main idea (as I understand it) is that $\pi \circ \sigma^{-1}$ takes the unit ball to $\mathbb{R}^{n-1}\times (-\infty, 0]$. It is the composition of continuous, open maps, and is thus continuous and open. You can show that, in a nbhd. of a boundary point (of the unit ball), it is injective, and thus defines a chart for each boundary point into (a space homeomorphic to) $\mathbb{H}^n$. $\endgroup$
    – Steve D
    Jan 24, 2018 at 22:53
  • $\begingroup$ @SteveD Thanks! I already solved this question (in the suggested manner) some time after posting it, but never bothered to post a solution. Half out of apathy, and half out of Prof. Lee's request in the intro to not post solutions. Thank you very much! $\endgroup$ Jan 24, 2018 at 22:55
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    $\begingroup$ Ah, OK, this question came up among several people today, and we could not find a nice exposition online, but only found your question! $\endgroup$
    – Steve D
    Jan 24, 2018 at 22:56
  • $\begingroup$ I struggled a bit with the hint because I'd come up with my own homeomorphism but I'd still like to understand how the hint works. I was initially very confused that the map just maps the half circle in B2 to another half circle, but the important detail that I missed is that the circular boundary is mapped to the line boundary, which is important to work with of H2 in the definition of manifold with boundary. $\endgroup$
    – cicolus
    Mar 26, 2022 at 11:42

1 Answer 1

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I've been worked on this problem for some time, and i think i probably solved it based on the hint given on the book. Maybe this seems a little long, but it is really not. I tried my best to make this simple. It is really helpful to follow the arguments with a picture for the case $n=2$ in mind.

It is enough to solve this for the closed unit ball $\bar{\mathbb{B}}^n$ in $\mathbb{R}^n$, since any other closed ball $\bar{B}_r(p)$ in $\mathbb{R}^n$ is homeomorphic to $\bar{\mathbb{B}}^n$ by composition of translation $T : \bar{B}_r(p) \to \bar{B}_r(0)$, defined as $x \mapsto x - p$ together with dilation $D : \bar{B}_r(0) \to \bar{\mathbb{B}}^n$, defined as $x \mapsto \frac{x}{r}$.

$\diamond \quad $ As a subspace of $\mathbb{R}^n$, $\bar{\mathbb{B}}^n$ is a second countable Hausdorff space. For any point $p\in \mathbb{B}^n$, the identity map on $\mathbb{B}^n$ serve as the homeomorphism. So we only need to construct homeomorphisms between neighbourhood of points on $\partial \bar{\mathbb{B}}^n=\mathbb{S}^{n-1}$ with open subsets in $\mathbb{H}^n$. To do this we need to consider $\bar{\mathbb{B}}^n$ as a subspace of $\mathbb{R}^{n+1}$.

Consider the stereographic projection from the south pole $\sigma : \mathbb{S}^n\smallsetminus \{S\} \to \mathbb{R}^n$, which is a homeomorphism, defined as $$ \sigma(x_1,\dots,x_{n+1}) = \frac{(x_1,\dots,x_n)}{1+x_{n+1}}. $$ For $i=1,\dots,n$, define $$U_i^{+} =\{ (x_1,\dots,x_n) \in \mathbb{R}^n : x_i > 0 \}, \quad U_i^{-} =\{ (x_1,\dots,x_n) \in \mathbb{R}^n : x_i < 0 \}$$ be $2n$-many open subsets of $\mathbb{R}^n$, and also for $i=1,\dots,n$ $$ \widetilde{U}_i^+ = \{(x_1,\dots,x_{n+1}) \in \mathbb{R}^{n+1} : x_i>0 \}, \quad \widetilde{U}_i^- = \{(x_1,\dots,x_{n+1}) \in \mathbb{R}^{n+1} : x_i<0 \} $$ are $2n$-many open subsets in $\mathbb{R}^{n+1}$.

Observe that $$\sigma^{-1}(U_i^{\pm}) = \mathbb{S}^n \cap \widetilde{U}_i^{\pm}$$ for each $i=1,\dots,n$. That is $\sigma^{-1}$ map $U_i^+$ to the open hemisphere of $\mathbb{S}^n$ where $x_i>0$, and same for $U_i^-$. In particular,

$\bullet \quad \sigma^{-1}$ is an identity map on $$\partial \bar{\mathbb{B}}^n = \mathbb{S}^{n-1} = \{(x_1,\dots,x_{n+1}) \in \mathbb{R}^{n+1} : x_{n+1} = 0 \text{ and }\sum_{i=1}^{n} x_i^2 = 1 \},$$

$\bullet \quad $ The inside region, $\mathbb{B}^n \cap U_i^+$, is maped to the upper part ($x_{n+1}>0$) of the hemisphere $\sigma^{-1}(U_i^+)$, and the outside region, $U_i^+ \smallsetminus \overline{\mathbb{B}^n}$ , is maped to the lower part ($x_{n+1}<0$) of hemisphere $\sigma^{-1}(U_i^+)$.

Since these hemispheres $\sigma^{-1}(U^{\pm}_i)$ homeomorphic to open unit ball $\mathbb{B}^{n}$ via projection map $\pi_i : (x_1,\dots,x_i,\dots,x_{n+1}) \mapsto (x_1,\dots,x_{i-1},x_{i+1},\dots,x_{n+1})$, then by restricting the composition map $\pi_i \circ \sigma^{-1} : \mathbb{R}^n \to \mathbb{R}^n$ to $U_i^{\pm} \cap \bar{\mathbb{B}}^n$, we obtain the desired homeomorphisms $$ \varphi:=\pi_i \circ (\sigma^{-1})|_{U_i^{\pm}\cap \bar{\mathbb{B}}^n} : U_i^{\pm}\cap \bar{\mathbb{B}}^n \to \mathbb{H}^n, $$ with domains cover $\partial\bar{\mathbb{B}}^n = \mathbb{S}^{n-1}$. By construction, any $p\in \mathbb{S}^{n-1}$ must contained in one such neighbourhoods, with $\varphi(p) \in \partial \mathbb{H}^n$ and $\varphi(U_i^{\pm}\cap \bar{\mathbb{B}}^n)$ is an open half unit ball in $\mathbb{H}^n$. Therefore, $\bar{\mathbb{B}}^n$ is an $n$-manifold with boundary with manifold boundary is equal to its topological boundary $\mathbb{S}^{n-1}$.

Note that by similar way we can show that the complement of any open ball is an $n$-manifold with boundary, with its topological boundary as the manifold boundary. Only this time we use stereographic projection form the north pole.

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  • $\begingroup$ Hey. My only question is how is the inverse image of the closed ball and the U plus or minus open under the inverse of the stenographic projection map? I can’t seem to make this inference to conclude. $\endgroup$
    – user82261
    Nov 19, 2019 at 5:43
  • $\begingroup$ @user82261 Stereographic projection is a homeomorphism. $\endgroup$ Nov 19, 2019 at 11:20
  • $\begingroup$ Right but the inverse image we get is $H^{n+1} \cap S^{n+1} \cap \tilde{U}^{\pm}$. This is open in the subspace topology in $H^{n+1} \cap S^{n+1}$ and not in general in the upper half plane. Or does if it not matter my? $\endgroup$
    – user82261
    Nov 19, 2019 at 15:01
  • $\begingroup$ Using the stereographic projection also shows that the closed unit ball is a smooth submanifold of $\mathbb{R}^n$. $\endgroup$ Dec 28, 2020 at 4:05

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