A-equivalence
In mathematics, [math]\displaystyle{ \mathcal{A} }[/math]-equivalence, sometimes called right-left equivalence, is an equivalence relation between map germs.
Let [math]\displaystyle{ M }[/math] and [math]\displaystyle{ N }[/math] be two manifolds, and let [math]\displaystyle{ f, g : (M,x) \to (N,y) }[/math] be two smooth map germs. We say that [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math] are [math]\displaystyle{ \mathcal{A} }[/math]-equivalent if there exist diffeomorphism germs [math]\displaystyle{ \phi : (M,x) \to (M,x) }[/math] and [math]\displaystyle{ \psi : (N,y) \to (N,y) }[/math] such that [math]\displaystyle{ \psi \circ f = g \circ \phi. }[/math]
In other words, two map germs are [math]\displaystyle{ \mathcal{A} }[/math]-equivalent if one can be taken onto the other by a diffeomorphic change of co-ordinates in the source (i.e. [math]\displaystyle{ M }[/math]) and the target (i.e. [math]\displaystyle{ N }[/math]).
Let [math]\displaystyle{ \Omega(M_x,N_y) }[/math] denote the space of smooth map germs [math]\displaystyle{ (M,x) \to (N,y). }[/math] Let [math]\displaystyle{ \mbox{diff}(M_x) }[/math] be the group of diffeomorphism germs [math]\displaystyle{ (M,x) \to (M,x) }[/math] and [math]\displaystyle{ \mbox{diff}(N_y) }[/math] be the group of diffeomorphism germs [math]\displaystyle{ (N,y) \to (N,y). }[/math] The group [math]\displaystyle{ G := \mbox{diff}(M_x) \times \mbox{diff}(N_y) }[/math] acts on [math]\displaystyle{ \Omega(M_x,N_y) }[/math] in the natural way: [math]\displaystyle{ (\phi,\psi) \cdot f = \psi^{-1} \circ f \circ \phi. }[/math] Under this action we see that the map germs [math]\displaystyle{ f, g : (M,x) \to (N,y) }[/math] are [math]\displaystyle{ \mathcal{A} }[/math]-equivalent if, and only if, [math]\displaystyle{ g }[/math] lies in the orbit of [math]\displaystyle{ f }[/math], i.e. [math]\displaystyle{ g \in \mbox{orb}_G(f) }[/math] (or vice versa).
A map germ is called stable if its orbit under the action of [math]\displaystyle{ G := \mbox{diff}(M_x) \times \mbox{diff}(N_y) }[/math] is open relative to the Whitney topology. Since [math]\displaystyle{ \Omega(M_x,N_y) }[/math] 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 [math]\displaystyle{ k }[/math]-jets for every [math]\displaystyle{ k }[/math] 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 [math]\displaystyle{ orb_G(f). }[/math] The map germ [math]\displaystyle{ f }[/math] 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 [math]\displaystyle{ (\mathbb{R}^n,0) \to (\mathbb{R},0) }[/math] for [math]\displaystyle{ 1 \le n \le 3 }[/math] are the infinite sequence [math]\displaystyle{ A_k }[/math] ([math]\displaystyle{ k \in \mathbb{N} }[/math]), the infinite sequence [math]\displaystyle{ D_{4+k} }[/math] ([math]\displaystyle{ k \in \mathbb{N} }[/math]), [math]\displaystyle{ E_6, }[/math] [math]\displaystyle{ E_7, }[/math] and [math]\displaystyle{ E_8. }[/math]
See also
- K-equivalence (contact equivalence)
References
- M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities. Graduate Texts in Mathematics, Springer.
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