= Embedding of the n-dimensional torus in codimension one
{title2=$T^n\hookrightarrow\mathbb R^{n+1}$}
Suppose a compact $m$-manifold $N$ is embedded in the half-space $x_{m+1}>0$ of $\mathbb R^{m+1}$. Spinning it around the boundary hyperplane gives the embedding
$$
N\times S^1\longrightarrow\mathbb R^{m+2},
\qquad
(x,e^{i\theta})\longmapsto
(x_1,\ldots,x_m,x_{m+1}\cos\theta,x_{m+1}\sin\theta).
$$
Starting with a circle and iterating proves that $T^n=(S^1)^n$ embeds in $\mathbb R^{n+1}$.
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