Difference between revisions of "Manifold is not union of images of manifolds of smaller dimension"
From Diffgeom
m (3 revisions) 

(No difference)

Latest revision as of 19:48, 18 May 2008
This fact is an application of the following pivotal fact/result/idea: Sard's theorem
View other applications of Sard's theorem OR Read a survey article on applying Sard's theorem
This article gives a purely topological (or settheoretic) constraint that we get when we restrict ourselves to the category of differential manifolds with smooth maps
Statement
Let be a smooth manifold and be a countable sequence of smooth manifolds, each having dimension strictly less than that of . Suppose are smooth maps. Then, is not the union of .
The analogous statement is not true for topological manifolds with continuous maps: there can be a continuous surjective map from a manifold of smaller dimension to a manifold of larger dimension.