Recipe for embedding $m$-fold product $X_m := S^1 \ \times ... \times \ S^1$ onto a bounded subset of $\mathbb{R}^{m+1}$

Question

I am an undergraduate student doing a first course in topology. I am stuck with a conclusion that I feel like should be relatively straightforward after proving quite some statements. The assignment is relatively long (giving a "recipe") for the final conclusion, so please bear with me.

Let $S^1$ denote the unit circle in $\mathbb{R}^2$. Consider for $n \geq 2$ the mapping:

$R: S^1 \rightarrow M_{n+1}$ given by $(u,v) \mapsto$ $ \begin{bmatrix} u & 0 & -v \\ 0 & I_{n-1} & 0 \\ v & 0 & u \end{bmatrix}$

I have proved the following things:

$(\textbf{A})$ that $R(\cos (\phi), \sin(\phi))$ defines a rotation by an angle $\phi$ in the $(x_{1}, x_{n+1})$-plane.

Now, letting $H:=\{x \in \mathbb{R}^n \vert x_1 >0 \}$

$(\textbf{B})$ that the map $f:H \times S^1 \mapsto \mathbb{R}^{n+1}$, $(x, u, v) \mapsto $ $R(u,v)\begin{bmatrix} x_{1} \\ x_{2} \\ \vdots \\ x_{n}\\ 0 \end{bmatrix}$ is a homeomorphism from

$H \times$ $S^1$ (equipped with the topology induced from $\mathbb{R}^n \times \mathbb{R}^2$) onto an open subset of $\mathbb{R}^{n+1}$.

$(\textbf{C})$ When $A \subset H$ and $A \times S^1 \subset \mathbb{R}^n \times \mathbb{R}^2$ are equipped with the induced topology, then $f|_{A×S^1}$ is an embedding of $A \times S^1$ onto $\mathbb{R}^{n+1}$.

$(\textbf{D})$Finally I showed that if $A$ is bounded in $\mathbb{R}^n$ that $f(A \times S^1)$ is bounded in $\mathbb{R}^{n+1}$.

Using these results I have to conclude the following:

For $m \geq$ 1 let the $m$-fold product $X_m := S^1 \ \times ... \times \ S^1 \subset (\mathbb{R}^2)^m = \mathbb{R}^{2m}$ be equipped with induced topology. Show for all $m \geq 1$ there exists an embedding of $X_m$ onto a bounded subset of $\mathbb{R}^{m+1}$ .

I feel like one can picture the $(m-1)$-fold product of $S^1$ to sit in $H$, as the bounded subset $A$ from part $\textbf{D}$. But I do not know how to realize this since $S^1$ is the unit circle and the first coordinate of elements of $H$ is greater than $0$.

Any help would be appreciated!

Answer 1

We prove that $X_m$ embeds into $\mathbb{R}^{m+1}$ as a bounded subset by induction on $m$.

The base case $m=1$ is just the statement that $S^1\subseteq \mathbb{R}^2$ is bounded.

Now assume inductively that $X_m$ embeds into $\mathbb{R}^{m+1}$ as a bounded subset. Let $g:X_m\rightarrow \mathbb{R}^{m+1}$ be such an embedding.

Because $g(X_m)$ is bounded it lies inside of some cube: $g(X_m)\subseteq [a_1, b_1]\times [a_2,b_2]\times...\times [a_{m+1}, b_{m+1}]$. Consider a new function $h:X_m\rightarrow \mathbb{R}^{m+1}$ given by $h(x) = g(x) + (2|a_1|,0,...,0)$. I leave it to you to show that $h$ is also an embedding and that $h(X_m)\subseteq H$.

Great. How does this give an embedding of $X_{m+1}$ into $\mathbb{R}^{m+2}$? Well, now that we have $h(X_m)\subseteq A$, part C can be used. Can you finish from here?