-equivalence, sometimes called right-left equivalence, is an equivalence relation between map germs.
be two smooth map germs.
-equivalent if there exist diffeomorphism germs
ϕ : (
ψ ∘ f = g ∘ ϕ .
In other words, two map germs are
-equivalent if one can be taken onto the other by a diffeomorphic change of co-ordinates in the source (i.e.
denote the space of smooth map germs
be the group of diffeomorphism germs
be the group of diffeomorphism germs
( ϕ , ψ ) ⋅ f =
∘ f ∘ ϕ .
Under this action we see that the map germs
lies in the orbit of
(or vice versa).
A map germ is called stable if its orbit under the action of
is open relative to the Whitney topology.
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
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
The map germ
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
are the infinite sequence
), the infinite sequence
This topology-related article is a stub.
You can help Wikipedia by expanding it.