Alexander trick

Alexander's trick, also known as the Alexander trick, is a basic result in geometric topology, named after J. W. Alexander.

Two homeomorphisms of the n-dimensional ball ${\displaystyle D^{n}}$ which agree on the boundary sphere ${\displaystyle S^{n-1}}$ are isotopic.

More generally, two homeomorphisms of Dⁿ that are isotopic on the boundary are isotopic.

Base case: every homeomorphism which fixes the boundary is isotopic to the identity relative to the boundary.

If ${\displaystyle f\colon D^{n}\to D^{n}}$ satisfies ${\displaystyle f(x)=x{\text{ for all }}x\in S^{n-1}}$, then an isotopy connecting f to the identity is given by

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