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
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 polynom ...
: 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
In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French '' bord'', giving ''cobordism'') of a manifold. Two manifolds of the same ...
, and the starting point for
surgery theory
In mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one finite-dimensional manifold from another in a 'controlled' way, introduced by . Milnor called this technique ''surgery'', while And ...
. 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
:
.
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
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ne ...
, then we can define the restriction of the map
to the boundary, as
. 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
:
.
Parametric transversality theorem
Consider the map
and define
. 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
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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