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 x , N y ) denote the space of smooth map germs ( M , x ) → ( N , y ) . Let diff ( M x ) be the group of diffeomorphism germs ( M , x ) → ( M , x ) and diff ( N y ) be the group of diffeomorphism germs ( N , y ) → ( N , y ) . The group G := diff ( M x ) × diff ( N y ) acts on Ω ( M x , N y ) in the natural way: ( ϕ , ψ ) ⋅ f = ψ − 1 ∘ 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 G ( f ) (or vice versa).
A map germ is called stable if its orbit under the action of G := diff ( M x ) × diff ( N y ) is open relative to the Whitney topology. Since Ω ( M x , N y ) 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 o r b G ( 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 n , 0 ) → ( R , 0 ) for 1 ≤ n ≤ 3 are the infinite sequence A k ( k ∈ N ), the infinite sequence D 4 + k ( k ∈ N ), E 6 , E 7 , and E 8 .
