Transversality theorem

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 $$f:X\rightarrow Y$$, may be deformed by an arbitrary small amount into a map that is transversal to a given submanifold $$Z \subseteq Y$$. 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.

Previous definitions
Let $$f:X\rightarrow Y$$ be a smooth map between manifolds, and let $$Z$$ be a submanifold of $$Y$$. We say that $$f$$ is transversal to $$Z$$, denoted as $$f \pitchfork Z$$, if and only if for every $$x\in f^{-1}\left(Z\right)$$ we have $$Im\left( df_x \right) + T_{f\left(x\right)} Z = T_{f\left(x\right)} Y$$.

An important result about transversality states that if a smooth map $$f$$ is transversal to $$Z$$, then $$f^{-1}\left(Z\right)$$ 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 $$\partial f:\partial X \rightarrow Y$$. The map $$\partial f$$ is smooth, and it allow us to state an extension of the previous result: if both $$f \pitchfork Z$$ and $$\partial f \pitchfork Z$$, then $$f^{-1}\left(Z\right)$$ is a regular submanifold of $$X$$ with boundary, and $$\partial f^{-1}\left( Z \right) = f^{-1}\left( Z \right) \cap \partial X$$.

The key to transversality is families of mappings. Consider the map $$F:X\times S \rightarrow Y$$ and define $$f_s\left(x\right) = F\left(x,s\right)$$. This generates a family of mappings $$f_s:X\rightarrow Y$$. We require that the family vary smoothly by assuming $$S$$ to be a manifold and $$F$$ to be smooth.

Formal statement
The formal statement of the transversality theorem is:

Suppose that $$F:X \times S \rightarrow Y$$ 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 $$\partial F$$ are transversal to $$Z$$, then for almost every $$s\in S$$, both $$f_s$$ and $$\partial f_s$$ are transversal to $$Z$$.

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 $$F:X \times S \rightarrow Y$$ is a $$C^k$$ map of $$C^\infty$$-Banach manifolds. Assume that

i- $$X$$, $$S$$ and $$Y$$ are nonempty, metrizable $$C^\infty$$-Banach manifols with chart spaces over a field $$\mathbb{K}$$.

ii- The $$C^k$$-map $$F:X \times S \rightarrow Y$$ with $$k\geq 1$$ has $$y$$ as a regular value.

iii- For each parameter $$s\in S$$, the map $$f_s\left(x\right) = F\left(x,s\right)$$ is a Fredholm map, where $$ind Df_s\left(x\right)<k$$ for every $$x\in f_{s}^{-1}\left( \left\{y\right\} \right)$$.

iv- The convergence $$s_n \rightarrow s$$ on $$S$$ as $$n \rightarrow \infty$$ and $$F\left(x_n, s_n \right) = y$$ for all $$n$$ implies the existence of a convergent subsequence $$x_n \rightarrow x$$ as $$n \rightarrow \infty$$ with $$x\in X$$.

If Assumptions i-iv hold, then there exists an open, dense subset $$S_0$$ of $$S$$ such that $$y$$ is a regular value of $$f_s$$ for each parameter $$s\in S_0$$.

Now, fix an element $$s\in S_0$$. If there exists a number $$n\geq 0$$ with $$ind Df_s\left( x \right) = n$$ for all solutions $$x\in X $$ of $$f_s\left(x \right) = y$$, then the solution set $$f_s^{-1}\left( \left\{y \right\} \right)$$ consists of an $$n$$-dimensional $$C^k$$-Banach manifold or the solution set is empty.

Note that if $$ind Df_s\left( x \right) = 0$$ for all the solutions of $$f_s\left(x \right) = y$$, then there exists an open dense subset $$S_0$$ of $$S$$ such that there are at most finitely many solutions for each fixed parameter $$s\in S_0$$. In addition, all these solutions are regular.