Isotopy of immersions: Difference between revisions

Revision as of 23:51, 3 April 2008

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

@@ Line 6: / Line 6: @@
 * <math>F(p,0) = f_0(p)</math> and <math>F(p,1) = f_1(p)</math>: In other words <math>F</math> is a homotopy from <math>f_0</math> to <math>f_1</math>
 * For each <math>t</math>, the map from <math>M</math> to <math>\N</math> given by <math>p \mapsto f(p,t)</math> is an immersion
+==Facts==
+* [[Immersions in Euclidean space are isotopic in its square]]