Transversality theorem

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In differential topology, the transversality theorem is a major result that describes the transversal 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 transversal to a given submanifold . The finite dimensional version of the transversality theorem is 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.

Contents 1 Finite dimensional version 1.1 Previous definitions 1.2 Formal statement 2 Infinite dimensional version 2.1 Formal statement 3 References

[edit] Finite dimensional version

[edit] Previous definitions

Let be a smooth map between manifolds, and let Z be a submanifold of Y. We say that f is transversal to Z, denoted as , if and only if for every we have .

An important result about transversality states that if a smooth map f is transversal to Z, then is a regular submanifold of X.

If X is a manifold with boundary, then we can define the restriction of the map f 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 X with boundary, and .

The key to transversality is families of mappings. Consider the map and define . This generates a family of mappings . We require that the family vary smoothly by assuming S to be a manifold and F to be smooth.

[edit] Formal statement

The formal statement of the transversality theorem is:

Suppose that is a smooth map of manifolds, where only X has boundary, and let Z be any submanifold of Y without boundary. If both F and are transversal to Z, then for almost every , both f_s and are transversal to Z.

[edit] Infinite dimensional version

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

[edit] Formal statement

Suppose that is a C^k map of -Banach manifolds. Assume that

i- X, S and Y are nonempty, metrizable -Banach manifols with chart spaces over a field .

ii- The C^k-map with has y as a regular value.

iii- For each parameter , the map is a Fredholm map, where for every .

iv- The convergence on S as and for all n implies the existence of a convergent subsequence as with .

If Assumptions i-iv hold, then there exists an open, dense subset S₀ of S such that y is a regular value of f_s for each parameter .

Now, fix an element . If there exists a number with for all solutions of , then the solution set consists of an n-dimensional C^k-Banach manifold or the solution set is empty.

Note that if for all the solutions of , then there exists an open dense subset S₀ of S such that there are at most finitely many solutions for each fixed parameter . In addition, all these solutions are regular.

[edit] References

• Guillemin, Victor and Pollack, Alan (1974) Differential Topology. Prentice-Hall. ISBN 0-13-212605-2.
• Zeidler, Eberhard (1997) Nonlinear Functional Analysis and Its Applications: Part 4: Applications to Mathematical Physics. Springer. ISBN 978-0387964997.