diffeotopy

Let $M$ be a manifold and $I=[0,1]$ the closed unit interval. A smooth map $h:M\times I\to M$ is called a diffeotopy (on $M$) if for every $t\in I$:

$$h_t:=h(-,t):M\to M$$

is a diffeomorphism.

Two diffeomorphisms $f,g:M\to M$ are said to be diffeotopic if there is a diffeotopy $h:M\times I\to M$ such that

1. $h_0=f$, and

2. $h_1=g$.

Remark. Diffeotopy is an equivalence relation among diffeomorphisms. In particular, those diffeomorphisms that are diffeotopic to the identity map form a group.

Two points $a,b\in M$ are said to be isotopic if there is a diffeotopy $h$ on $M$ such that

1. $h_0=i{d}_M$, the identity map on $M$, and

2. $h_1(a)=b$.

Remark. If $M$ is a connected manifold, then every pair of points on $M$ are isotopic.

Pairs of isotopic points in a manifold can be generazlied to pairs of isotopic sets. Two arbitrary sets $A,B\subseteq M$ are said to be isotopic if there is a diffeotopy $h$ on $M$ such that

1. $h_0=i{d}_M$, and

2. $h_1(A)=B$.

Remark. One special example of isotopic sets is the isotopy of curves. In ${ℝ}^3$, curves that are isotopic to the unit circle are the trivial knots.