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.

Finite dimensional version

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 $indDf_{s}\left(x\right)<k$ 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 $indDf_{s}\left(x\right)=n$ 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 $indDf_{s}\left(x\right)=0$ 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.

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.