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 ${\mathbb{R}}^{3}$, curves that are isotopic to the unit circle are the trivial knots^{}.
Title  diffeotopy 

Canonical name  Diffeotopy 
Date of creation  20130322 14:52:43 
Last modified on  20130322 14:52:43 
Owner  rspuzio (6075) 
Last modified by  rspuzio (6075) 
Numerical id  9 
Author  rspuzio (6075) 
Entry type  Definition 
Classification  msc 57R50 
Defines  isotopic 
Defines  diffeotopic 