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

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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:

$\map {f^{-1} } x$ is a submanifold of $M$ with boundary.

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

Then $\map {f^{-1} } x$ is one dimensional and compact.

Since $\partial f$ is the identity mapping:

$\map {\partial f^{-1} } x = \map {f^{-1} } x \cap \partial M = \set x$

This contradicts the Classification of Compact One-Manifolds.

$\blacksquare$