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. 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

be a submanifold, if and only if we have that

1. .

.

is a manifold with boundary with boundary, and

1. .

Parametric transversality theorem

to be smooth.

The statement of the parametric transversality theorem is:

.

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 Gδ 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

-Banach manifolds. Assume:

1. as a regular value.

   is a Fredholm map

   implies the existence of a convergent subsequence

-Banach manifold or the solution set is empty.

In addition, all these solutions are regular.

References

Bibliography