A-equivalence

In mathematics, ${𝒜}$-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 ${𝒜}$-equivalent if there exist diffeomorphism germs $\varphi :(M,x)\to (M,x)$ and $\psi :(N,y)\to (N,y)$ such that $\psi \circ f=g\circ \varphi .$ In other words, two map germs are ${𝒜}$-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 $\mathrm{\Omega }(M_x,N_y)$ denote the space of smooth map germs $(M,x)\to (N,y).$ Let $\text{diff}(M_x)$ be the group of diffeomorphism germs $(M,x)\to (M,x)$ and $\text{diff}(N_y)$ be the group of diffeomorphism germs $(N,y)\to (N,y).$ The group $G:=\text{diff}(M_x)\times \text{diff}(N_y)$ acts on $\mathrm{\Omega }(M_x,N_y)$ in the natural way: $(\varphi ,\psi )\cdot f={\psi }^{-1}\circ f\circ \varphi .$ Under this action we see that the map germs $f,g:(M,x)\to (N,y)$ are ${𝒜}$-equivalent if, and only if, $g$ lies in the orbit of $f$, i.e. $g\in {\text{orb}}_G(f)$ (or vice versa). A map germ is called stable if its orbit under the action of $G:=\text{diff}(M_x)\times \text{diff}(N_y)$ is open relative to the Whitney topology. Since $\mathrm{\Omega }(M_x,N_y)$ 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 $or{b}_G(f).$ 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 $A_k$ ($k\in \mathbb{\mathbb{N}}$), the infinite sequence $D_{4+k}$ ($k\in \mathbb{\mathbb{N}}$), $E_6,$ $E_7,$ and $E_8.$

See also

• K-equivalence (contact equivalence)

References

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