Regular Homotopies in the Plane Part 1 - Listen - Topology Films

Regular Homotopies in the Plane Part 1

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Description

This video defines the meaning of a regular curve (a smooth closed curve) in the plane, and the rules for a regular homotopy between regular curves. It then defines the rotation number of a regular curve, and proves that there is a regular homotopy between two regular curves only if they have the same rotation number. This proof constitutes the first half of the Whitney Graustein theorem (see Hassler Whitney, "On regular closed curves in the plane" Compositio Mathematica, 4 (1937), p. 276–284.)

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This video completes the proof of Whitney Graustein theorem, by showing that if two regular curves have the same rotation number, then there is a regularly homotopy between them. The construction of the regular homotopy proceeds by the same algorithm that was used to produce the computer...

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