Transversality theorem
Encyclopedia
In differential topology
, the transversality theorem, also known as the Thom
Transversality Theorem, is a major result that describes the transversal intersection properties of a smooth family of smooth maps. It says that transversality is a generic property
: any smooth map
, may be deformed by an arbitrary small amount into a map that is transversal to a given submanifold 
. Together with the Pontryagin-Thom construction, it is the technical heart of cobordism theory, and the starting point for surgery theory
. The finite dimensional version of the transversality theorem is also a very useful tool for establishing the genericity of a property which is dependent on a finite number of real parameters and which is expressible using a system of nonlinear equations. This can be extended to an infinite dimensional parametrization using the infinite dimensional version of the transversality theorem.
be a smooth map between manifolds, and let 
be a submanifold of 
. We say that 
is transversal to 
, denoted as 
, if and only if for every 
we have 
.
An important result about transversality states that if a smooth map
is transversal to 
, then 
is a regular submanifold of 
.
If
is a manifold with boundary, then we can define the restriction of the map 
to the boundary, as 
. The map 
is smooth, and it allow us to state an extension of the previous result: if both 
and 
, then 
is a regular submanifold of 
with boundary, and 
.
The key to transversality is families of mappings. Consider the map
and define 
. This generates a family of mappings 
. We require that the family vary smoothly by assuming 
to be a manifold and 
to be smooth.
Suppose that
is a smooth map of manifolds, where only 
has boundary, and let 
be any submanifold of 
without boundary. If both 
and 
are transversal to 
, then for almost every 
, both 
and 
are transversal to 
.
is a 
map of 
-Banach manifolds. Assume that
i-
, 
and 
are nonempty, metrizable 
-Banach manifols with chart spaces over a field 
.
ii- The
-map 
with 
has 
as a regular value.
iii- For each parameter
, the map 
is a Fredholm map
, where
for every 
.
iv- The convergence
on 
as 
and 
for all 
implies the existence of a convergent subsequence 
as 
with 
.
If Assumptions i-iv hold, then there exists an open, dense subset
of 
such that 
is a regular value of 
for each parameter 
.
Now, fix an element
. If there exists a number 
with 
for all solutions 
of 
, then the solution set 
consists of an 
-dimensional 
-Banach manifold or the solution set is empty.
Note that if
for all the solutions of 
, then there exists an open dense subset 
of 
such that there are at most finitely many solutions for each fixed parameter 
. In addition, all these solutions are regular.
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
, the transversality theorem, also known as the Thom
René Thom
René Frédéric Thom was a French mathematician. He made his reputation as a topologist, moving on to aspects of what would be called singularity theory; he became world-famous among the wider academic community and the educated general public for one aspect of this latter interest, his work as...
Transversality Theorem, is a major result that describes the transversal intersection properties of a smooth family of smooth maps. It says that transversality is a generic property
Generic property
In mathematics, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functions, as in the statements, "A generic polynomial does not have a root at zero," or "A generic...
: any smooth map
, may be deformed by an arbitrary small amount into a map that is transversal to a given submanifold . Together with the Pontryagin-Thom construction, it is the technical heart of cobordism theory, and the starting point for surgery theorySurgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along...
. The finite dimensional version of the transversality theorem is also a very useful tool for establishing the genericity of a property which is dependent on a finite number of real parameters and which is expressible using a system of nonlinear equations. This can be extended to an infinite dimensional parametrization using the infinite dimensional version of the transversality theorem.
Previous definitions
Let be a smooth map between manifolds, and let be a submanifold of . We say that is transversal to , denoted as , if and only if for every we have .An important result about transversality states that if a smooth map
is transversal to , then is a regular submanifold of .If
is a manifold with boundary, then we can define the restriction of the map to the boundary, as . The map is smooth, and it allow us to state an extension of the previous result: if both and , then is a regular submanifold of with boundary, and .The key to transversality is families of mappings. Consider the map
and define . This generates a family of mappings . We require that the family vary smoothly by assuming to be a manifold and to be smooth.Formal statement
The formal statement of the transversality theorem is:Suppose that
is a smooth map of manifolds, where only has boundary, and let be any submanifold of without boundary. If both and are transversal to , then for almost every , both and are transversal to .Infinite dimensional version
The infinite dimensional version of the transversality theorem takes into account that the manifolds may be modeled in Banach spaces.Formal statement
Suppose that is a map of -Banach manifolds. Assume thati-
, and are nonempty, metrizable -Banach manifols with chart spaces over a field .ii- The
-map with has as a regular value.iii- For each parameter
, the map is a Fredholm mapFredholm operator
In mathematics, a Fredholm operator is an operator that arises in the Fredholm theory of integral equations. It is named in honour of Erik Ivar Fredholm....
, where
for every .iv- The convergence
on as and for all implies the existence of a convergent subsequence as with .If Assumptions i-iv hold, then there exists an open, dense subset
of such that is a regular value of for each parameter .Now, fix an element
. If there exists a number with for all solutions of , then the solution set consists of an -dimensional -Banach manifold or the solution set is empty.Note that if
for all the solutions of , then there exists an open dense subset of such that there are at most finitely many solutions for each fixed parameter . In addition, all these solutions are regular.