Isotopy

Two embeddings $f,g:N\to\Rr^m$$<wikitex> Two embeddings $f,g:N\to\Rr^m$ are said to be $(ambient)$ isotopic (see \cite{Skopenkov2006}, Figure 1.1), if there exists a homeomorphism onto (an ambient isotopy) $F:\Rr^m\times I\to\Rr^m\times I$ such that * $F(y,0)=(y,0)$ for each $y\in\Rr^m,$ * $F(f(x),1)=(g(x),1)$ for each $x\in N,$ and * $F(\Rr^m\times\{t\})=\Rr^m\times\{t\}$ for each $t\in I.$ An ambient isotopy is also a homotopy $\Rr^m\times I\to\Rr^m$ or a family of homeomorphisms $F_t:\Rr^m\to\Rr^m$ generated by the map $F$ in the obvious manner. Evidently, isotopy is an equivalence relation on the set of embeddings of $N$ into $\Rr^m$. </wikitex>f,g:N\to\Rr^m$ are said to be $(ambient)$$(ambient)$ isotopic (see [Skopenkov2006], Figure 1.1), if there exists a homeomorphism onto (an ambient isotopy) $F:\Rr^m\times I\to\Rr^m\times I$$F:\Rr^m\times I\to\Rr^m\times I$ such that

• $F(y,0)=(y,0)$$F(y,0)=(y,0)$ for each $y\in\Rr^m,$$y\in\Rr^m,$
• $F(f(x),1)=(g(x),1)$$F(f(x),1)=(g(x),1)$ for each $x\in N,$$x\in N,$ and
• $F(\Rr^m\times\{t\})=\Rr^m\times\{t\}$$F(\Rr^m\times\{t\})=\Rr^m\times\{t\}$ for each $t\in I.$$t\in I.$

An ambient isotopy is also a homotopy $\Rr^m\times I\to\Rr^m$$\Rr^m\times I\to\Rr^m$ or a family of homeomorphisms $F_t:\Rr^m\to\Rr^m$$F_t:\Rr^m\to\Rr^m$ generated by the map $F$$F$ in the obvious manner.

Evidently, isotopy is an equivalence relation on the set of embeddings of $N$$N$ into $\Rr^m$$\Rr^m$.