Transversality theorem explained

In differential topology, the transversality theorem, also known as the Thom transversality theorem after French mathematician René Thom, is a major result that describes the transverse intersection properties of a smooth family of smooth maps. It says that transversality is a generic property: any smooth map

f\colonXY

, may be deformed by an arbitrary small amount into a map that is transverse to a given submanifold

Z\subseteqY

. Together with the Pontryagin–Thom construction, it is the technical heart of cobordism theory, and the starting point for surgery theory. The finite-dimensional version of the transversality theorem is also 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\colonXY

be a smooth map between smooth manifolds, and let

Z

be a submanifold of

Y

. We say that

f

is transverse to

Z

, denoted as

f\pitchforkZ

, if and only if for every

x\inf-1\left(Z\right)

we have that

\operatorname{im}\left(dfx\right)+Tf\left(x\right)Z=Tf\left(x\right)Y

.

An important result about transversality states that if a smooth map

f

is transverse 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

\partialf\colon\partialXY

. The map

\partialf

is smooth, and it allows us to state an extension of the previous result: if both

f\pitchforkZ

and

\partialf\pitchforkZ

, then

f-1\left(Z\right)

is a regular submanifold of

X

with boundary, and

\partialf-1\left(Z\right)=f-1\left(Z\right)\cap\partialX

.

Parametric transversality theorem

Consider the map

F\colonX x SY

and define

fs\left(x\right)=F\left(x,s\right)

. This generates a family of mappings

fs\colonXY

. We require that the family vary smoothly by assuming

S

to be a (smooth) manifold and

F

to be smooth.

The statement of the parametric transversality theorem is:

Suppose that

F\colonX x SY

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

\partialF

are transverse to

Z

, then for almost every

s\inS

, both

fs

and

\partialfs

are transverse to

Z

.

More general transversality theorems

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

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.

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

F:X x S\toY

is a

Ck

map of

Cinfty

-Banach manifolds. Assume:

(i)

X,S

and

Y

are non-empty, metrizable

Cinfty

-Banach manifolds with chart spaces over a field

K.

(ii) The

Ck

-map

F:X x S\toY

with

k\geq1

has

y

as a regular value.

(iii) For each parameter

s\inS

, the map

fs(x)=F(x,s)

is a Fredholm map, where

\operatorname{ind}Dfs(x)<k

for every

x\in

-1
f
s

(\{y\}).

(iv) The convergence

sn\tos

on

S

as

n\toinfty

and

F(xn,sn)=y

for all

n

implies the existence of a convergent subsequence

xn\tox

as

n\toinfty

with

x\inX.

If (i)-(iv) hold, then there exists an open, dense subset

S0\subsetS

such that

y

is a regular value of

fs

for each parameter

s\inS0.

Now, fix an element

s\inS0.

If there exists a number

n\geq0

with

\operatorname{ind}Dfs(x)=n

for all solutions

x\inX

of

fs(x)=y

, then the solution set
-1
f
s

(\{y\})

consists of an

n

-dimensional

Ck

-Banach manifold or the solution set is empty.

Note that if

\operatorname{ind}Dfs(x)=0

for all the solutions of

fs(x)=y,

then there exists an open dense subset

S0

of

S

such that there are at most finitely many solutions for each fixed parameter

s\inS0.

In addition, all these solutions are regular.

References