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 $F : M \times I \to R^{n}$ 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 $N$ 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]]