The Fundamental Theorem on Universal Unfoldings

The second major theorem in Catastrophe Theory is René Thorn's Fundamental Theorem on Universal Unfoldings, which will be proved in this chapter. In addition to its theoretical importance, this theorem gives a concrete method for calculating a universal unfolding of a given germ. An important application is Table 1 of Chapter 7, which lists the universal unfoldings of the 7 elementary catastrophes. The proof of the Fundamental Theorem on Universal Unfoldings relies heavily on the so-called Main Lemma, which will be treated first. After that, just relatively simple algebraic arguments will be needed to derive the Fundamental Theorem. The proof of the Main Lemma is essentially based on the same idea used in (4.22), which was the major tool to show Mather's Sufficient Criterion for Determinacy (4.24). In particular, the problem is reduced to solving an ordinary differential equation.

Main Lemma (1). Let k and r be positive integers, and let [f] be a k-determined germ in https://www.w3.org/1998/Math/MathML"> m n 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429501807/0363dac7-c66e-432e-878a-a354faa10a85/content/inequ10_197_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> . Then any two r-parameter k-transversal unfoldings of [f] are equivalent.

Proof. The proof will be subdivided into six parts. For every r-parameter unfolding [H], let γ _j(H) be the germ in m_n of the function https://www.w3.org/1998/Math/MathML"> ∂ ∂ u i H ( x , 0 ) − ∂ ∂ u j H ( 0 , 0 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429501807/0363dac7-c66e-432e-878a-a354faa10a85/content/unequ10_197_1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>

defined for small x in R ⁿ ; recall that the γ _j(H), j = l,…,r, span the space https://www.w3.org/1998/Math/MathML"> V [ H ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429501807/0363dac7-c66e-432e-878a-a354faa10a85/content/inequ10_197_2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> introduced in Chapter 8.

(i) There is an unfolding [H] in ℰ _n +r of [f] such that for every k-transversal unfolding [F] of [f] in ℰ _{n + r} there exists a k-transversal unfolding [F′] of [f] in ℰ _{n + r} equivalent to [H] and satisfying https://www.w3.org/1998/Math/MathML"> γ j ( F ′ ) − γ j ( F ) ∈ J [ f ] https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429501807/0363dac7-c66e-432e-878a-a354faa10a85/content/inequ10_197_3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> for j = l,…,r.