No-retraction theorem

Statement
The fact about::sphere $$S^n = \partial D^{n+1}$$ is not a fact about::retract of the fact about::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