No-retraction theorem: Difference between revisions

Revision as of 04:09, 24 December 2010

This article describes a theorem about spheres

Statement

The sphere $S^{n}=\partial D^{n+1}$ is not a retract of the disk $D^{n+1}$ . In other words, the sphere is not a retract of the disk that it bounds.

Equivalently, the identity map from $S^{n}$ to itself is not nullhomotopic, and hence $S^{n}$ is not contractible.

Corollaries

Complex numbers are algebraically closed uses the two-dimensional case of this theorem
Brouwer fixed-point theorem

@@ Line 3: / Line 3: @@
 ==Statement==
-The sphere <math>S^n</math> is ''not'' a retract of the disc <math>D^{n+1}</math>. In other words, the sphere is not a retract of the disc that it bounds.
+The sphere <math>S^n = \partial D^{n+1}</math> is ''not'' a [[retract]] of the disk <math>D^{n+1}</math>. In other words, the sphere is not a retract of the disk that it bounds.
 Equivalently, the identity map from <math>S^n</math> to itself is not [[nullhomotopic map|nullhomotopic]], and hence <math>S^n</math> is not contractible.