A-equivalence

In mathematics, ${\displaystyle {\mathcal {A}}}$-equivalence, sometimes called right-left equivalence, is an equivalence relation between map germs.

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

In other words, two map germs are ${\displaystyle {\mathcal {A}}}$-equivalent if one can be taken onto the other by a diffeomorphic change of co-ordinates in the source (i.e. ${\displaystyle M}$) and the target (i.e. ${\displaystyle N}$).

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

A map germ is called stable if its orbit under the action of ${\displaystyle G:={\mbox{diff}}(M_{x})\times {\mbox{diff}}(N_{y})}$ is open relative to the Whitney topology. Since ${\displaystyle \Omega (M_{x},N_{y})}$ 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 ${\displaystyle k}$-jets for every ${\displaystyle k}$ 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 ${\displaystyle orb_{G}(f).}$ The map germ ${\displaystyle f}$ 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 ${\displaystyle (\mathbb {R} ^{n},0)\to (\mathbb {R} ,0)}$ for ${\displaystyle 1\leq n\leq 3}$ are the infinite sequence ${\displaystyle A_{k}}$ (${\displaystyle k\in \mathbb {N} }$), the infinite sequence ${\displaystyle D_{4+k}}$ (${\displaystyle k\in \mathbb {N} }$), ${\displaystyle E_{6},}$ ${\displaystyle E_{7},}$ and ${\displaystyle E_{8}.}$