A-equivalence

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In mathematics, \mathcal{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) \to (N,y) be two smooth map germs. We say that f and g are \mathcal{A}-equivalent if there exist diffeomorphism germs \phi : (M,x) \to (M,x) and \psi : (N,y) \to (N,y) such that \psi \circ f = g \circ \phi.

In other words, two map germs are \mathcal{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 \Omega(M_x,N_y) denote the space of smooth map germs (M,x) \to (N,y). Let \mbox{diff}(M_x) be the group of diffeomorphism germs (M,x) \to (M,x) and \mbox{diff}(N_y) be the group of diffeomorphism germs (N,y) \to (N,y). The group  G := \mbox{diff}(M_x) \times \mbox{diff}(N_y) acts on \Omega(M_x,N_y) in the natural way:  (\phi,\psi) \cdot f = \psi^{-1} \circ f \circ \phi. Under this action we see that the map germs f, g : (M,x) \to (N,y) are \mathcal{A}-equivalent if, and only if, g lies in the orbit of f, i.e.  g \in \mbox{orb}_G(f) (or vice versa).

A map germ is called stable if its orbit under the action of  G := \mbox{diff}(M_x) \times \mbox{diff}(N_y) is open relative to the Whitney topology. Since \Omega(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 orb_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 (\mathbb{R}^n,0) \to (\mathbb{R},0) for 1 \le n \le 3 are the infinite sequence A_k (k \in \mathbb{N}), the infinite sequence D_{4+k} (k \in \mathbb{N}), E_6, E_7, and E_8.

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References[edit]

  • M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities. Graduate Texts in Mathematics, Springer.