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)\to (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)\to (M,x)}
  
 and 
  
    
      
        ψ
        :
        (
        N
        ,
        y
        )
        →
        (
        N
        ,
        y
        )
      
    
    {\displaystyle \psi :(N,y)\to (N,y)}
  
 such that 
  
    
      
        ψ
        ∘
        f
        =
        g
        ∘
        ϕ
        .
      
    
    {\displaystyle \psi \circ f=g\circ \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)\to (N,y).}
  
 Let 
  
    
      
        
          
            diff
          
        
        (
        
          M
          
            x
          
        
        )
      
    
    {\displaystyle {\mbox{diff}}(M_{x})}
  
 be the group of diffeomorphism germs 
  
    
      
        (
        M
        ,
        x
        )
        →
        (
        M
        ,
        x
        )
      
    
    {\displaystyle (M,x)\to (M,x)}
  
 and 

  
    
      
        
          
            diff
          
        
        (
        
          N
          
            y
          
        
        )
      
    
    {\displaystyle {\mbox{diff}}(N_{y})}
  
 be the group of diffeomorphism germs 
  
    
      
        (
        N
        ,
        y
        )
        →
        (
        N
        ,
        y
        )
        .
      
    
    {\displaystyle (N,y)\to (N,y).}
  
 The group 
  
    
      
        G
        :=
        
          
            diff
          
        
        (
        
          M
          
            x
          
        
        )
        ×
        
          
            diff
          
        
        (
        
          N
          
            y
          
        
        )
      
    
    {\displaystyle G:={\mbox{diff}}(M_{x})\times {\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 f\circ \phi .}
  
 Under this action we see that the map germs 
  
    
      
        f
        ,
        g
        :
        (
        M
        ,
        x
        )
        →
        (
        N
        ,
        y
        )
      
    
    {\displaystyle f,g:(M,x)\to (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 g\in {\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})\times {\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 of
these 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)\to (\mathbb {R} ,0)}
  
 for 
  
    
      
        1
        ≤
        n
        ≤
        3
      
    
    {\displaystyle 1\leq n\leq 3}
  
 are the infinite sequence 
  
    
      
        
          A
          
            k
          
        
      
    
    {\displaystyle A_{k}}
  
 (
  
    
      
        k
        ∈
        
          N
        
      
    
    {\displaystyle k\in \mathbb {N} }
  
), the infinite sequence 
  
    
      
        
          D
          
            4
            +
            k
          
        
      
    
    {\displaystyle D_{4+k}}
  
 (
  
    
      
        k
        ∈
        
          N
        
      
    
    {\displaystyle k\in \mathbb {N} }
  
), 
  
    
      
        
          E
          
            6
          
        
        ,
      
    
    {\displaystyle E_{6},}
  
 
  
    
      
        
          E
          
            7
          
        
        ,
      
    
    {\displaystyle E_{7},}
  
 and 
  
    
      
        
          E
          
            8
          
        
        .
      
    
    {\displaystyle E_{8}.}
  

 
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