Isotopy of immersions

Definition

Suppose $M,N$ are smooth manifolds of dimensions $m,n$ and $f_{0},f_{1}:M\to N$ are two smooth immersions. An isotopy from $f_{0}$ to $f_{1}$ is a map $F:M\times I\to \mathbb {R} ^{n}$ such that the following hold:

• $F$ is a smooth map
• $F(p,0)=f_{0}(p)$ and $F(p,1)=f_{1}(p)$: In other words $F$ is a homotopy from $f_{0}$ to $f_{1}$
• For each $t$, the map from $M$ to $\mathbb {N}$ given by $p\mapsto f(p,t)$ is an immersion

Facts

• Immersions in Euclidean space are isotopic in its square