Retraction Theorem

Theorem
Let $M$ be a compact manifold with boundary $\partial M$.

Then there is no smooth mapping $f: M \to \partial M$ such that $\partial f: \partial M \to \partial M$ is the identity.

Proof
Aiming for a contradiction, suppose such a smooth mapping exists.

By the Morse-Sard Theorem, there exists a regular value $x \in \partial M$.

By the Preimage Theorem:
 * $f^{-1} \left({x}\right)$ is a submanifold of $M$ with boundary.

We have that the codimension of $f^{-1} \left({x}\right)$ in $M$ equals the codimension of $x$ in $\partial M$, that is, $\dim \left({M}\right) - 1$.

Then $f^{-1} \left({x}\right)$ is one dimensional and compact.

Since $\partial f$ is the identity mapping:
 * $\partial f^{-1} \left({x}\right) = f^{-1} \left({x}\right) \cap \partial M = \left\{ {x}\right\}$

This contradicts the Classification of Compact One-Manifolds.