Transversality theorem

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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

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Previous definitions

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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, 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

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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

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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

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The infinite-dimensional version of the transversality theorem takes into account that the manifolds may be modeled in Banach spaces. [citation needed]

Formal statement

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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   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   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.

References

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  • Arnold, Vladimir I. (1988). Geometrical Methods in the Theory of Ordinary Differential Equations. Springer. ISBN 0-387-96649-8.
  • Golubitsky, Martin; Guillemin, Victor (1974). Stable Mappings and Their Singularities. Springer-Verlag. ISBN 0-387-90073-X.
  • Guillemin, Victor; Pollack, Alan (1974). Differential Topology. Prentice-Hall. ISBN 0-13-212605-2.
  • Hirsch, Morris W. (1976). Differential Topology. Springer. ISBN 0-387-90148-5.
  • Thom, René (1954). "Quelques propriétés globales des variétés differentiables". Commentarii Mathematici Helvetici. 28 (1): 17–86. doi:10.1007/BF02566923.
  • Thom, René (1956). "Un lemme sur les applications différentiables". Bol. Soc. Mat. Mexicana. 2 (1): 59–71.
  • Zeidler, Eberhard (1997). Nonlinear Functional Analysis and Its Applications: Part 4: Applications to Mathematical Physics. Springer. ISBN 0-387-96499-1.