Transversality theorem explained
In differential topology, the transversality theorem, also known as the Thom transversality theorem after French mathematician René Thom, is a major result that describes the transverse 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 transverse 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.
Finite-dimensional version
Previous definitions
Let
be a smooth map between smooth manifolds, and let
be a submanifold of
. We say that
is transverse to
, denoted as
, if and only if for every
we have that
\operatorname{im}\left(dfx\right)+Tf\left(x\right)Z=Tf\left(x\right)Y
.
An important result about transversality states that if a smooth map
is transverse 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
\partialf\colon\partialX → Y
. The map
is smooth, and it allows us to state an extension of the previous result: if both
and
, then
is a regular submanifold of
with boundary, and
\partialf-1\left(Z\right)=f-1\left(Z\right)\cap\partialX
.
Parametric transversality theorem
Consider the map
and define
fs\left(x\right)=F\left(x,s\right)
. This generates a family of mappings
. We require that the family vary smoothly by assuming
to be a (smooth) manifold and
to be smooth.
The statement of the parametric 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 transverse to
, then for almost every
, both
and
are transverse to
.
More general transversality theorems
The parametric transversality theorem above is sufficient for many elementary applications (see the book by Guillemin and Pollack).
There are more powerful statements (collectively known as transversality theorems) that imply the parametric transversality theorem and are needed for more advanced applications.
Informally, the "transversality theorem" states that the set of mappings that are transverse to a given submanifold is a dense open (or, in some cases, only a dense
) subset of the set of mappings. To make such a statement precise, it is necessary to define the space of mappings under consideration, and what is the topology in it. There are several possibilities; see the book by Hirsch.
What is usually understood by Thom's transversality theorem is a more powerful statement about jet transversality. See the books by Hirsch and by Golubitsky and Guillemin. The original reference is Thom, Bol. Soc. Mat. Mexicana (2) 1 (1956), pp. 59–71.
John Mather proved in the 1970s an even more general result called the multijet transversality theorem. See the book by Golubitsky and Guillemin.
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
is a
map of
-Banach manifolds. Assume:
(i)
and
are non-empty, metrizable
-Banach manifolds 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
\operatorname{ind}Dfs(x)<k
for every
(iv) The convergence
on
as
and
for all
implies the existence of a convergent subsequence
as
with
If (i)-(iv) hold, then there exists an open, dense subset
such that
is a regular value of
for each parameter
Now, fix an element
If there exists a number
with
\operatorname{ind}Dfs(x)=n
for all solutions
of
, then the solution set
consists of an
-dimensional
-Banach manifold or the solution set is empty.
Note that if
\operatorname{ind}Dfs(x)=0
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.
References
- Book: Arnold, Vladimir I. . Vladimir Arnold . Geometrical Methods in the Theory of Ordinary Differential Equations . Springer . 1988 . 0-387-96649-8 . registration .
- Book: Marty Golubitsky . Martin . Golubitsky . Victor . Guillemin . Stable Mappings and Their Singularities . Springer-Verlag . 1974 . 0-387-90073-X .
- Book: Guillemin, Victor . Victor Guillemin. Pollack . Alan . 1974 . Differential Topology . Prentice-Hall . 0-13-212605-2 .
- Book: Hirsch, Morris W. . Morris Hirsch . Differential Topology . Springer . 1976 . 0-387-90148-5 .
- René . Thom . René Thom. Quelques propriétés globales des variétés differentiables . . 28 . 1954 . 1 . 17–86 . 10.1007/BF02566923 .
- René . Thom . Un lemme sur les applications différentiables . Bol. Soc. Mat. Mexicana . 2 . 1 . 1956 . 59–71 .
- Book: Zeidler, Eberhard . 1997 . Nonlinear Functional Analysis and Its Applications: Part 4: Applications to Mathematical Physics . Springer . 0-387-96499-1 .