A-equivalence

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 $$E_8.$$