Transversality theorem

Introduction

In differential topology, the transversality theorem is a main 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.

References

• Guillemin, Victor and Pollack, Alan (1974) Differential Topology. Prentice-Hall. ISBN 0-13-212605-2.