Balaban 11-cage

In mathematics, and in particular singularity theory an A_k, where k ≥ 0 is an integer, describes a level of degeneracy of a function. The notation was introduced by V. I. Arnold.

Let f : Rⁿ → R be a smooth function. We denote by Ω(Rⁿ,R) the infinite-dimensional space of all such functions. Let diff(Rⁿ) denote the infinite-dimensional Lie group of diffeomorphisms Rⁿ → Rⁿ, and diff(R) the infinite-dimensional Lie group of diffeomorphisms R → R. The product group diff(Rⁿ) × diff(R) acts on Ω(Rⁿ,R) in the following way: let φ : Rⁿ → Rⁿ and ψ : R → R be diffeormorphisms and f : Rⁿ → R any smooth function. We define the group action as follows:

$(\varphi ,\psi )\cdot f:=\psi \circ f\circ \varphi ^{-1}$

The orbit of f, denoted orb(f), of this group action is given by

${\mbox{orb}}(f)=\{\psi \circ f\circ \varphi ^{-1}:\varphi \in {\mbox{diff}}({\mathbf {R} }^{n}),\psi \in {\mbox{diff}}({\mathbf {R} })\}\ .$

The members of a given orbit of this action have the following fact in common: we can find a diffeomorphic change of coordinate in Rⁿ and a diffeomorphic change of coordinate in R such that one member of the orbit is carried to any other. A function f is said to have a type A_k-singularity if it lies in the orbit of

$f(x_{1},\ldots ,x_{n})=1+\varepsilon _{1}x_{1}^{2}+\cdots +\varepsilon _{n-1}x_{n-1}^{2}\pm x_{n}^{k+1}$

where $\varepsilon _{i}=\pm 1$ and k ≥ 0 is an integer.

By a normal form we mean a particularly simple representative of any given orbit. The above expressions for f give normal forms for the type A_k-singularities. The type A_k-singularities are special because they are amongst the simple singularities, this means that there are only a finite number of other orbits in a sufficiently small neighbourhood of the orbit of f.

This idea extends over the complex numbers where the normal forms are much simpler; for example: there is no need to distinguish ε_i = +1 from ε_i = −1.

References

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