Pseudo-isotopy

In mathematics , pseudo-isotopy is a generalization of the concept of isotopy .

Various problems of differential topology can be traced back to the question of whether pseudo-isotopic images are also isotopic.

definition

Two diffeomorphisms

${\displaystyle f_{0},f_{1}\colon M\to M}$

a differentiable manifold is called pseudo-isotopic if there is a diffeomorphism

${\displaystyle F\colon M\times \left[0,1\right]\to M\times \left[0,1\right]}$

are, the limitation on or with or matches. ${\displaystyle M\times \left\{0\right\}}$ ${\displaystyle M\times \left\{1\right\}}$ ${\displaystyle f_{0}}$ ${\displaystyle f_{1}}$

A pseudo-isotope is an isotope when the level set is mapped to itself for all . ${\displaystyle F}$ ${\displaystyle t\in \left[0,1\right]}$ ${\displaystyle M\times \left\{t\right\}}$

Cerf's pseudoisotopy theorem

Cerf's pseudoisotopy theorem is a generalization of the h-cobordism theorem .

It says that for all simply connected manifolds of dimension two pseudo-isotopic mappings are always isotopic. ${\displaystyle n\geq 5}$ ${\displaystyle f_{0},f_{1}}$

For manifolds that are not simply connected, however, there are obstructions in the algebraic K-theory .

application

From the pseudoisotopy theorem it follows that there is a bijection between the exotic spheres in dimension and the connected components of the diffeomorphism group . ${\displaystyle n\geq 5}$ ${\displaystyle n+1}$ ${\displaystyle S^{n}}$

literature

• Jean Cerf : La stratification naturelle des espaces de fonctions differentiables réelles et le théoreme de la pseudo-isotopie . Publ. IHES 39, 5-173 (1970).