Embedding torus in Euclidean space

Question

For $n > 2$, is it possible to embed $\underbrace{S^1 \times \cdots \times S^1}_{n\text{ times}}$ into $\mathbb R^{n+1}$?

Answer 1

As Aaron Mazel-Gee's comment indicates, this follows from induction. Although you only asked about $n > 2$, it actually holds for $n \geq 1$.

The base case is $n = 1$, i.e. $S^1$ embeds in $\mathbb{R}^2$, which is clear

For the inductive step, suppose that $T^{k-1}$ embeds in $\mathbb{R}^k$. Then $T^k = T^{k-1}\times S^1$ embeds in $\mathbb{R}^k\times S^1$ which is diffeomorphic to $\mathbb{R}^{k-1}\times(\mathbb{R}\times S^1) \cong \mathbb{R}^{k-1}\times(\mathbb{R}^2\setminus\{(0, 0)\})$. Now note that $\mathbb{R}^{k-1}\times(\mathbb{R}^2\setminus\{(0, 0)\})$ embeds into $\mathbb{R}^{k-1}\times\mathbb{R}^2 \cong \mathbb{R}^{k+1}$. As the composition of embeddings is again an embedding, we see that $T^k$ embeds in $\mathbb{R}^{k+1}$.

By induction, $T^n$ embeds in $\mathbb{R}^{n+1}$ for every $n \geq 1$.

Answer 2

Let $e_0, \ldots, e_n$ be the standard basis of $\mathbb {R}^{n+1}$. Take $\epsilon$ small. Consider the vector $v_1$ of length 1 in the span of $e_0, e_1$. Then the vector $v_2$ of length $\epsilon$ in the span of $v_1, e_2$, and in general the vector $v_i$ of length $\epsilon^{i-1}$ in the span of $v_{i-1}, e_i$. Now consider the vector $w=v_1+\ldots v_n$ For small $\epsilon$ the set of $w$'s is a torus embedded $\mathbb {R}^{n+1}$ (any $\epsilon < 1$ will do, actually).