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