Transversality theorem

Let ${\displaystyle f\colon X\rightarrow Y}$ be a smooth map between smooth manifolds, and let ${\displaystyle Z}$ be a submanifold of ${\displaystyle Y}$. We say that ${\displaystyle f}$ is transverse to ${\displaystyle Z}$, denoted as ${\displaystyle f\pitchfork Z}$, if and only if for every ${\displaystyle x\in f^{-1}\left(Z\right)}$ we have that

${\displaystyle \operatorname {im} \left(df_{x}\right)+T_{f\left(x\right)}Z=T_{f\left(x\right)}Y}$.

An important result about transversality states that if a smooth map ${\displaystyle f}$ is transverse to ${\displaystyle Z}$, then ${\displaystyle f^{-1}\left(Z\right)}$ is a regular submanifold of ${\displaystyle X}$.

If ${\displaystyle X}$ is a manifold with boundary, then we can define the restriction of the map ${\displaystyle f}$ to the boundary, as ${\displaystyle \partial f\colon \partial X\rightarrow Y}$. The map ${\displaystyle \partial f}$ is smooth, and it allows us to state an extension of the previous result: if both ${\displaystyle f\pitchfork Z}$ and ${\displaystyle \partial f\pitchfork Z}$, then ${\displaystyle f^{-1}\left(Z\right)}$ is a regular submanifold of ${\displaystyle X}$ with boundary, and

${\displaystyle \partial f^{-1}\left(Z\right)=f^{-1}\left(Z\right)\cap \partial X}$.

Consider the map ${\displaystyle F\colon X\times S\rightarrow Y}$ and define ${\displaystyle f_{s}\left(x\right)=F\left(x,s\right)}$. This generates a family of mappings ${\displaystyle f_{s}\colon X\rightarrow Y}$. We require that the family vary smoothly by assuming ${\displaystyle S}$ to be a (smooth) manifold and ${\displaystyle F}$ to be smooth.

The statement of the parametric transversality theorem is:

Suppose that ${\displaystyle F\colon X\times S\rightarrow Y}$ is a smooth map of manifolds, where only ${\displaystyle X}$ has boundary, and let ${\displaystyle Z}$ be any submanifold of ${\displaystyle Y}$ without boundary. If both ${\displaystyle F}$ and ${\displaystyle \partial F}$ are transverse to ${\displaystyle Z}$, then for almost every ${\displaystyle s\in S}$, both ${\displaystyle f_{s}}$ and ${\displaystyle \partial f_{s}}$ are transverse to ${\displaystyle Z}$.

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 ${\displaystyle G_{\delta }}$) 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.

Suppose ${\displaystyle F:X\times S\to Y}$ is a ${\displaystyle C^{k}}$ map of ${\displaystyle C^{\infty }}$-Banach manifolds. Assume:

(i) ${\displaystyle X,S}$ and ${\displaystyle Y}$ are non-empty, metrizable ${\displaystyle C^{\infty }}$-Banach manifolds with chart spaces over a field ${\displaystyle \mathbb {K} .}$

(ii) The ${\displaystyle C^{k}}$-map ${\displaystyle F:X\times S\to Y}$ with ${\displaystyle k\geq 1}$ has ${\displaystyle y}$ as a regular value.

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

(iv) The convergence ${\displaystyle s_{n}\to s}$ on ${\displaystyle S}$ as ${\displaystyle n\to \infty }$ and ${\displaystyle F(x_{n},s_{n})=y}$ for all ${\displaystyle n}$ implies the existence of a convergent subsequence ${\displaystyle x_{n}\to x}$ as ${\displaystyle n\to \infty }$ with ${\displaystyle x\in X.}$

If (i)-(iv) hold, then there exists an open, dense subset ${\displaystyle S_{0}\subset S}$ such that ${\displaystyle y}$ is a regular value of ${\displaystyle f_{s}}$ for each parameter ${\displaystyle s\in S_{0}.}$

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

Note that if ${\displaystyle \operatorname {ind} Df_{s}(x)=0}$ for all the solutions of ${\displaystyle f_{s}(x)=y,}$ then there exists an open dense subset ${\displaystyle S_{0}}$ of ${\displaystyle S}$ such that there are at most finitely many solutions for each fixed parameter ${\displaystyle s\in S_{0}.}$ In addition, all these solutions are regular.