In mathematics, an unfolding of a function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

 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 mapping (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

Function space

In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...

  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

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 of diffeomorphism

Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...

s of etc, acts

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

 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

Foliation

In mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....

  - 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

Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

 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

Quotient ring

In ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...

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.