Math 2.0

Manifolds and cell complexes Theorem

Whitney's theorem on folds and cusps

Builds on Singularity of a smooth mapSmooth manifoldWhitney C-infinity topologyDiffeomorphism

Statement

Let MM be a smooth surface, and give C∞(M,ℝ2)C^\infty(M, \R^2) the Whitney C∞C^\infty topology. The smooth maps f ⁣:M→ℝ2f \colon M \to \R^2 with the following property form a dense subset of C∞(M,ℝ2)C^\infty(M, \R^2). At every point p∈Mp \in M, there are local coordinates centred at pp and at f(p)f(p) in which ff has one of these forms:

  1. (x,y)↦(x,y)(x, y) \mapsto (x, y), a regular point;
  2. (x,y)↦(x,y2)(x, y) \mapsto (x, y^2), a fold point;
  3. (x,y)↦(x,y3+xy)(x, y) \mapsto (x, y^3 + x y), a cusp point.

Remarks

At a fold point the singular set is a smooth curve and the kernel of the differential is transverse to it. At a cusp point the kernel is tangent to the singular curve. For the cusp map f(x,y)=(x,y3+xy)f(x, y) = (x, y^3 + xy) the singular set is the parabola x=−3y2x = -3y^2, and its image is the cusp curve (−3y2,−2y3)(-3y^2, -2y^3), which satisfies 27v2=−4u327 v^2 = -4 u^3 in the target coordinates (u,v)(u, v). The proof uses transversality of the jet of ff to the subsets of the jet space defined by these singularities. Worse singularities have codimension greater than 22, so generic maps of a surface avoid them. Both the fold and the cusp are stable germs, so they cannot be removed by small perturbations. The theorem is due to Whitney and was a starting point of the theory of stable maps.

Going deeper

For a student

This is the first nontrivial case of the classification of generic singularities of maps between manifolds of equal dimension. In the language of Thom–Boardman singularities, folds are the Σ1,0\Sigma^{1,0} points and cusps are the Σ1,1,0\Sigma^{1,1,0} points. Higher-dimensional analogues are the Morin singularities, and the fold and cusp reappear in catastrophe theory as the fold and cusp catastrophes.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptSingularity of a smooth map, concept, no evidence levelSingularity of a smoo…conceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptWhitney C-infinity topology, concept, no evidence levelWhitney C-infinity to…conceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptDiffeomorphism, concept, no evidence levelDiffeomorphismconceptWhitney's theorem on folds and cusps, statement, no evidence levelWhitney's theorem on …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. Euclidean space object
  2. Function concept
  3. Singularity of a smooth map concept
  4. Smooth manifold concept
  5. Whitney C-infinity topology concept
  6. Total derivative concept
  7. Function of several variables concept
  8. Differentiable function of several variables concept
  9. Diffeomorphism concept
  10. Whitney's theorem on folds and cusps statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Singularity_theory
About this page
TagSSWFTA4GP8
URIm2://SSWFTA4GP8
Kindstatement, theorem
WritingW0 stub
MSC57R45, 58K25, 58K40
StatusThis copy of the site has no build data. Run m2 status SSWFTA4GP8.
AboutSingularity of a smooth map
UsesSingularity of a smooth map, Smooth manifold, Whitney C-infinity topology, Diffeomorphism

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