Math 2.0

Manifolds and cell complexes Theorem

Whitney–Graustein theorem

Builds on Regular curveClosed curveWinding numberHomotopy

Statement

Let a regular closed curve be a 11-periodic map γ ⁣:ℝ→ℝ2\gamma \colon \R \to \R^2 of class C1C^1 with γ′(t)≠0\gamma'(t) \ne 0 for all tt. Its turning number is the degree of the map t↦γ′(t)/∥γ′(t)∥t \mapsto \gamma'(t) / \norm{\gamma'(t)} from ℝ/ℤ\R / \Z to the unit circle. A regular homotopy between γ0\gamma_0 and γ1\gamma_1 is a continuous family (γu)u∈[0,1](\gamma_u)_{u \in [0, 1]} of regular closed curves such that (u,t)↦γu′(t)(u, t) \mapsto \gamma_u'(t) is continuous. Two regular closed curves in the plane are regularly homotopic if and only if they have the same turning number.

Remarks

The turning number is the winding number of the derivative γ′\gamma' around 00. It is invariant under regular homotopy, because it is an integer that depends continuously on uu. For the converse, one reparametrises both curves proportionally to arc length. Their unit tangent maps have the same degree, so they are joined by a homotopy of maps to the circle. Integrating these tangent fields and correcting the closing condition gives the required family. Example: the circle t↦(cos⁡2πt,sin⁡2πt)t \mapsto (\cos 2\pi t, \sin 2\pi t) has turning number 11, and the figure eight t↦(sin⁡4πt,sin⁡2πt)t \mapsto (\sin 4\pi t, \sin 2\pi t) has turning number 00. So these two curves are homotopic as maps but not regularly homotopic. The theorem was proved by Hassler Whitney in 1937, who credited the result to William Graustein.

Going deeper

For a student

The theorem classifies immersions of the circle into the plane up to regular homotopy by the integer turning number. Smale later extended this classification to immersions of spheres, which led to the eversion of the sphere. For a regular closed curve of class C2C^2, the turning number equals the total signed curvature divided by 2π2\pi.

Reading path

To understand this page from the ground up, read these 22 pages in order. The foundations come first.

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Complex numbers, object, no evidence levelComplex numbersobjectEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptHomotopy, concept, no evidence levelHomotopyconceptContour integral, definition, no evidence levelContour integraldefinitionFréchet derivative, definition, no evidence levelFréchet derivativedefinitionFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptWinding number, definition, no evidence levelWinding numberdefinitionLimit of a function of several variables, concept, no evidence levelLimit of a function o…conceptContinuity in R^n, definition, no evidence levelContinuity in R^ndefinitionDifferentiable function of several variables, definition, no evidence levelDifferentiable functi…definitionPartial derivative, definition, no evidence levelPartial derivativedefinitionDifferentiable on an open set, definition, no evidence levelDifferentiable on an …definitionHigher partial derivatives, definition, no evidence levelHigher partial deriva…definitionIterated derivative, definition, no evidence levelIterated derivativedefinitionC^k function characterised by continuous derivatives, statement, no evidence levelC^k function characte…statementParametrised curve, definition, no evidence levelParametrised curvedefinitionClosed curve, definition, no evidence levelClosed curvedefinitionTangent vector of a curve, definition, no evidence levelTangent vector of a c…definitionRegular curve, definition, no evidence levelRegular curvedefinitionWhitney–Graustein theorem, statement, no evidence levelWhitney–Graustein the…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Complex numbers object
  2. Euclidean space object
  3. Function concept
  4. Homotopy concept
  5. Contour integral definition
  6. Fréchet derivative definition
  7. Function of several variables concept
  8. Vector-valued function concept
  9. Winding number definition
  10. Limit of a function of several variables concept
  11. Continuity in R^n definition
  12. Differentiable function of several variables definition
  13. Partial derivative definition
  14. Differentiable on an open set definition
  15. Higher partial derivatives definition
  16. Iterated derivative definition
  17. C^k function characterised by continuous derivatives statement
  18. Parametrised curve definition
  19. Closed curve definition
  20. Tangent vector of a curve definition
  21. Regular curve definition
  22. Whitney–Graustein theorem statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Regular_homotopy
About this page
TagZR6DEE6RBC
URIm2://ZR6DEE6RBC
Kindstatement, theorem
WritingW0 stub
MSC57R42, 53A04
StatusThis copy of the site has no build data. Run m2 status ZR6DEE6RBC.
AboutRegular curve
UsesRegular curve, Closed curve, Winding number, Homotopy
Changed2 November 2026

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