A-equivalence

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

Let M and N be two manifolds, and let be two smooth map germs. We say that f and g are -equivalent if there exist diffeomorphism germs and such that

In other words, two map germs are -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 Let diff(M_x) be the group of diffeomorphism germs and diff(N_y) be the group of diffeomorphism germs The group acts on Ω(M_x,N_y) in the natural way: Under this action we see that the map germs are -equivalent if, and only if, g lies in the orbit of f, i.e. (or visa-versa).

A map germ is called stable if its orbit under the action of 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 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 for are the infinite sequence A_k (), the infinite sequence D_{4 + k} (), E₆, E₇, and E₈.