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

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

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

This contradicts the Classification of Compact One-Manifolds.

$\blacksquare$