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.

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.