No-retraction theorem

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