TITLE: Embedding a torus in $\mathbb{R}^{n+1}$
QUESTION [9 upvotes]: It is easy to see that $T^2=S^1 \times  S^1$ can be embedded in $\mathbb{R}^4$ but also there is an embedded in $\mathbb{R}^3$. 
The question is $T^n = S^1 \times \ldots \times S^1$ can be embedded into $\mathbb{R}^{n+1}$, $n>2$.

REPLY [11 votes]: If we have an embedding $T^k\subset\mathbb{R}^{k+1}$ then the subspace inclusion $\mathbb{R}^{k+1}\subset\mathbb{R}^{k+2}$ allows us to view $T^k$ as embedded in $\mathbb{R}^{k+2}$.  The boundary of the tubular neighborhood of $T^k$ in $\mathbb{R}^{k+2}$ will then be a torus $T^{k+1}$ embedded in $\mathbb{R}^{k+2}$, and we proceed inductively.