A-equivalence

In mathematics, A {displaystyle {mathcal {A}}} -equivalence, sometimes called right-left equivalence, is an equivalence relation between map germs. In mathematics, A {displaystyle {mathcal {A}}} -equivalence, sometimes called right-left equivalence, is an equivalence relation between map germs. Let M {displaystyle M} and N {displaystyle N} be two manifolds, and let f , g : ( M , x ) → ( N , y ) {displaystyle f,g:(M,x) o (N,y)} be two smooth map germs. We say that f {displaystyle f} and g {displaystyle g} are A {displaystyle {mathcal {A}}} -equivalent if there exist diffeomorphism germs ϕ : ( M , x ) → ( M , x ) {displaystyle phi :(M,x) o (M,x)} and ψ : ( N , y ) → ( N , y ) {displaystyle psi :(N,y) o (N,y)} such that ψ ∘ f = g ∘ ϕ . {displaystyle psi circ f=gcirc phi .} In other words, two map germs are A {displaystyle {mathcal {A}}} -equivalent if one can be taken onto the other by a diffeomorphic change of co-ordinates in the source (i.e. M {displaystyle M} ) and the target (i.e. N {displaystyle N} ). Let Ω ( M x , N y ) {displaystyle Omega (M_{x},N_{y})} denote the space of smooth map germs ( M , x ) → ( N , y ) . {displaystyle (M,x) o (N,y).} Let diff ( M x ) {displaystyle {mbox{diff}}(M_{x})} be the group of diffeomorphism germs ( M , x ) → ( M , x ) {displaystyle (M,x) o (M,x)} and diff ( N y ) {displaystyle {mbox{diff}}(N_{y})} be the group of diffeomorphism germs ( N , y ) → ( N , y ) . {displaystyle (N,y) o (N,y).} The group G := diff ( M x ) × diff ( N y ) {displaystyle G:={mbox{diff}}(M_{x}) imes {mbox{diff}}(N_{y})} acts on Ω ( M x , N y ) {displaystyle Omega (M_{x},N_{y})} in the natural way: ( ϕ , ψ ) ⋅ f = ψ − 1 ∘ f ∘ ϕ . {displaystyle (phi ,psi )cdot f=psi ^{-1}circ fcirc phi .} Under this action we see that the map germs f , g : ( M , x ) → ( N , y ) {displaystyle f,g:(M,x) o (N,y)} are A {displaystyle {mathcal {A}}} -equivalent if, and only if, g {displaystyle g} lies in the orbit of f {displaystyle f} , i.e. g ∈ orb G ( f ) {displaystyle gin {mbox{orb}}_{G}(f)} (or vice versa). A map germ is called stable if its orbit under the action of G := diff ( M x ) × diff ( N y ) {displaystyle G:={mbox{diff}}(M_{x}) imes {mbox{diff}}(N_{y})} is open relative to the Whitney topology. Since Ω ( M x , N y ) {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 k {displaystyle k} -jets for every k {displaystyle 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 o r b G ( f ) . {displaystyle orb_{G}(f).} The map germ f {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 ( R n , 0 ) → ( R , 0 ) {displaystyle (mathbb {R} ^{n},0) o (mathbb {R} ,0)} for 1 ≤ n ≤ 3 {displaystyle 1leq nleq 3} are the infinite sequence A k {displaystyle A_{k}} ( k ∈ N {displaystyle kin mathbb {N} } ), the infinite sequence D 4 + k {displaystyle D_{4+k}} ( k ∈ N {displaystyle kin mathbb {N} } ), E 6 , {displaystyle E_{6},} E 7 , {displaystyle E_{7},} and E 8 . {displaystyle E_{8}.}