What is "a-equivalence"

In mathematics, A-equivalence, sometimes called right-left equivalence, is an equivalence relation between map germs.

Let M and N be two manifolds, and let f, g : (M, x) → (N, y) be two smooth map germs. We say that f and g are A-equivalent if there exist diffeomorphism germs ϕ : (M, x) → (M, x) and ψ : (N, y) → (N, y) such that ψ ∘ f = g ∘ ϕ.

In other words, two map germs are A-equivalent if one can be taken onto the other by a diffeomorphic change of co-ordinates in the source (i.e. M) and the target (i.e. N).

Let Ω(M, N) denote the space of smooth map germs (M, x) → (N, y). Let diff(M) be the group of diffeomorphism germs (M, x) → (M, x) and diff(N) be the group of diffeomorphism germs (N, y) → (N, y). The group G :  = diff(M) × diff(N) acts on Ω(M, N) in the natural way: (ϕ, ψ) ⋅ f = ψ ∘ f ∘ ϕ. Under this action we see that the map germs f, g : (M, x) → (N, y) are A-equivalent if, and only if, g lies in the orbit of f, i.e. g ∈ orb(f) (or vice versa).

A map germ is called stable if its orbit under the action of G :  = diff(M) × diff(N) is open relative to the Whitney topology. Since Ω(M, N) is an infinite dimensional space metric topology is no longer trivial. Whitney topology compares the differences in successive derivatives and gives a notion of proximity within the infinite dimensional space. A base for the open sets of the topology in question is given by taking k-jets for every k and taking open neighbourhoods in the ordinary Euclidean sense. Open sets in the topology are then unions of these base sets.

Consider the orbit of some map germ orb(f). The map germ f is called simple if there are only finitely many other orbits in a neighbourhood of each of its points. Vladimir Arnold has shown that the only simple singular map germs (R, 0) → (R, 0) for 1 ≤ n ≤ 3 are the infinite sequence A (k ∈ N), the infinite sequence D (k ∈ N), E,  E,  and E.