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 $f,\; g\; :\; (M,x)\; o\; (N,y)$ be two smooth map germs. We say that $f$ and $g$ are $mathcal\{A\}$-equivalent if there exist diffeomorphism germs $phi\; :\; (M,x)\; o\; (M,x)$ and $psi\; :\; (N,y)\; o\; (N,y)$ 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 $(M,x)\; o\; (N,y).$ Let $mbox\{diff\}(M\_x)$ be the group of diffeomorphism germs $(M,x)\; o\; (M,x)$ and $mbox\{diff\}(N\_y)$ be the group of diffeomorphism germs $(N,y)\; o\; (N,y).$ The group $G\; :=\; mbox\{diff\}(M\_x)\; imes\; mbox\{diff\}(N\_y)$ 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 $f,\; g\; :\; (M,x)\; o\; (N,y)$ are $mathcal\{A\}$-equivalent if, and only if, $g$ lies in the orbit of $f$, i.e. $g\; in\; mbox\{orb\}\_G(f)$ (or visa-versa).

A map germ is called stable if its orbit under the action of $G\; :=\; mbox\{diff\}(M\_x)\; imes\; mbox\{diff\}(N\_y)$ 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 ofthese 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 $(mathbb\{R\}^n,0)\; o\; (mathbb\{R\},0)$ 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 $E\_8.$

ee also

* K-equivalence (contact equivalence)

References

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