No-retraction theorem: Difference between revisions

Latest revision as of 04:10, 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 Closed unit 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

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 ==Statement==
-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.
+The [[fact about::sphere]] <math>S^n = \partial D^{n+1}</math> is ''not'' a [[fact about::retract]] of the [[fact about::closed unit 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.