Unfolding (functions)
In mathematics, an unfolding of a function is a certain family of functions.
Let be a smooth manifold and consider a smooth mapping
Let us assume that for given
and
we have
. Let
be a smooth
-dimensional manifold, and consider the family of mappings (parameterised by
) given by
We say that
is a
-parameter unfolding of
if
for all
In other words the functions
and
are the same: the function
is contained in, or is unfolded by, the family
Let be given by
An example of an unfolding of
would be
given by
As is the case with unfoldings, and
are called variables and
and
are called parameters – since they parameterise the unfolding.
In practice we require that the unfoldings have certain nice properties. In notice that
is a smooth mapping from
to
and so belongs to the function space
As we vary the parameters of the unfolding we get different elements of the function space. Thus, the unfolding induces a function
The space
where
denotes the group of diffeomorphisms of
etc., acts on
The action is given by
If
lies in the orbit of
under this action then there is a diffeomorphic change of coordinates in
and
which takes
to
(and vice versa). One nice property that we may like to impose is that
where "" denotes "transverse to". This property ensures that as we vary the unfolding parameters we can predict – by knowing how the orbit foliate
– how the resulting functions will vary.
There is an idea of a versal unfolding. Every versal unfolding has the property that
, but the converse is false. Let
be local coordinates on
, and let
denote the ring of smooth functions. We define the Jacobian ideal of
denoted by
as follows:
Then a basis for a versal unfolding of is given by quotient
This quotient is known as the local algebra of The dimension of the local algebra is called the Milnor number of
. The minimum number of unfolding parameters for a versal unfolding is equal to the Milnor number; that is not to say that every unfolding with that many parameters will be versal! Consider the function
A calculation shows that
This means that give a basis for a versal unfolding, and that
is a versal unfolding. A versal unfolding with the minimum possible number of unfolding parameters is called a miniversal unfolding.
Sometimes unfoldings are called deformations, versal unfoldings are called versal deformations, etc.
An important object associated to an unfolding is its bifurcation set. This set lives in the parameter space of the unfolding, and gives all parameter values for which the resulting function has degenerate singularities.
References
- V. I. Arnold, S. M. Gussein-Zade & A. N. Varchenko, Singularities of differentiable maps, Volume 1, Birkhäuser, (1985).
- J. W. Bruce & P. J. Giblin, Curves & singularities, second edition, Cambridge University press, (1992).