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
)
→
(
N
,
y
)
be two smooth map germs. We say that
f
and
g
are
A
-equivalent if there exist diffeomorphism germs
ϕ
:
(
M
,
x
)
→
(
M
,
x
)
and
ψ
:
(
N
,
y
)
→
(
N
,
y
)
such that
ψ
∘
f
=
g
∘
ϕ
.
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
Ω
(
M
x
,
N
y
)
denote the space of smooth map germs
(
M
,
x
)
→
(
N
,
y
)
.
Let
diff
(
M
x
)
be the group of diffeomorphism germs
(
M
,
x
)
→
(
M
,
x
)
and
diff
(
N
y
)
be the group of diffeomorphism germs
(
N
,
y
)
→
(
N
,
y
)
.
The group
G
:=
diff
(
M
x
)
×
diff
(
N
y
)
acts on
Ω
(
M
x
,
N
y
)
in the natural way:
(
ϕ
,
ψ
)
⋅
f
=
ψ
−
1
∘
f
∘
ϕ
.
Under this action we see that the map germs
f
,
g
:
(
M
,
x
)
→
(
N
,
y
)
are
A
-equivalent if, and only if,
g
lies in the orbit of
f
, i.e.
g
∈
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
)
is open relative to the Whitney topology. Since
Ω
(
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
o
r
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
(
R
n
,
0
)
→
(
R
,
0
)
for
1
≤
n
≤
3
are the infinite sequence
A
k
(
k
∈
N
), the infinite sequence
D
4
+
k
(
k
∈
N
),
E
6
,
E
7
,
and
E
8
.
