Whitney Greystone's theorem

The set of Whitney Gray Stone is a theorem from differential topology . He classifies curves in the plane using the tangent revolution number introduced by Carl Friedrich Gauß .

It is named after Hassler Whitney and William Caspar Graustein .

Curves in the plane

A closed regular curve in the plane is an illustration with for everyone . Two regular curves are called regular homotopic if there is a homotopy between them that is a regular curve at all times. ${\ displaystyle f \ colon S ^ {1} \ to \ mathbb {R} ^ {2}}$ ${\ displaystyle f ^ {\ prime} (x) \ not = (0,0)}$ ${\ displaystyle x}$

The number of revolutions of a curve in relation to a point represents the number of turns in counterclockwise direction when following the course of the curve. A turn around in clockwise direction results in the negative number of turns −1. ${\ displaystyle \ gamma}$ ${\ displaystyle z_ {0}}$ ${\ displaystyle z_ {0}}$

Number of turns
1	−1	0	1	2

The tangent revolution number of a regular curve is the revolution number of the tangent as a mapping in relation to the zero point . ${\ displaystyle S ^ {1} \ to \ mathbb {R} ^ {2} \ setminus \ left \ {(0,0) \ right \}}$ ${\ displaystyle (0,0)}$

Whitney Greystone's theorem

Whitney-Graustein's theorem states that closed regular curves in the plane are regularly homotopic if and only if they have the same tangent revolution number.

Generalizations

Smale generalized this theorem to curves in higher-dimensional manifolds and more generally to immersions of spheres: for two immersions are regular homotopic if and only if their obstruction classes in the homotopy group of the Stiefel manifold match. ${\ displaystyle k \ leq n-1}$ ${\ displaystyle S ^ {k} \ to \ mathbb {R} ^ {n}}$ ${\ displaystyle \ tau (f) \ in \ pi _ {k} V_ {n, k + 1}}$

literature

H. Whitney : On regular curves in the plane , Compos. Math. 4, 276–284, 1937. numdam (pdf)
K. Mehlhorn , C.-K. Yap : Constructive Whitney-Graustein Theorem: or how to untangle closed planar curves , SIAM J. Comput. 20, 603-621, 1991.

Web links

Whitney-Graustein Theorem (Encyclopedia of Mathematics)
Whitney-Graustein Theorem (MathWorld)

Individual evidence

↑ Whitney, Compositio Math., Volume 4, 1937, p. 279, writes that the theorem with proof was brought to his attention by Graustein

[1] Whitney, Compositio Math., Volume 4, 1937, p. 279, writes that the theorem with proof was brought to his attention by Graustein