# Transversality theorem

In differential topology, the **transversality theorem**, also known as the Thom Transversality Theorem, 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 manifolds, and let be a submanifold of . We say that is transverse to , denoted as , if and only if for every we have .

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 allow 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 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 that is a map of -Banach manifolds. Assume that

i) , and are nonempty, 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 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.

## References

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