Retraction Theorem

Theorem
If $M$ is some compact manifold with boundary, then there is no smooth map $f:M \to \partial M$ such that $\partial f: \partial M \to \partial M$ is the identity.

Proof
Suppose such a map exists, and let $x \in \partial M$ be a regular value, which must exist due to Sard's Theorem.

Then $f^{-1}(x)$ is a submanifold of $M$ with boundary, by the Preimage Theorem.

Since the codimension of $f^{-1}(x)$ in $M$ equals the codimension of $x$ in $\partial M$, that is, $\dim(M) - 1, f^{-1}(x)$ is one dimensional and compact.

Since $\partial f = $ the identity, $\partial f^{-1}(x) = f^{-1}(x) \cap \partial M = \left\{{ x }\right\}$.

This contradicts the Classification of Compact One-Manifolds.