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 $1\leq n\leq 3$ 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 $E_{8}.$