Submersion (mathematics) explained

In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential pushforward is everywhere surjective. It is a basic concept in differential topology, dual to that of an immersion.

Definition

Let M and N be differentiable manifolds, and let

f\colonM\toN

be a differentiable map between them. The map is a submersion at a point

p\inM

if its differential

Dfp\colonTpM\toTf(p)N

is a surjective linear map.[1] In this case, is called a regular point of the map ; otherwise, is a critical point. A point

q\inN

is a regular value of if all points in the preimage

f-1(q)

are regular points. A differentiable map that is a submersion at each point

p\inM

is called a submersion. Equivalently, is a submersion if its differential

Dfp

has constant rank equal to the dimension of .

Some authors use the term critical point to describe a point where the rank of the Jacobian matrix of at is not maximal.:[2] Indeed, this is the more useful notion in singularity theory. If the dimension of is greater than or equal to the dimension of, then these two notions of critical point coincide. However, if the dimension of is less than the dimension of, all points are critical according to the definition above (the differential cannot be surjective), but the rank of the Jacobian may still be maximal (if it is equal to dim). The definition given above is the more commonly used one, e.g., in the formulation of Sard's theorem.

Submersion theorem

Given a submersion

f\colonM\toN

between smooth manifolds of dimensions

m

and

n

, for each

x\inM

there exist surjective charts

\phi:U\toRm

of

M

around

x

, and

\psi:V\toRn

of

N

around

f(x)

, such that

f

restricts to a submersion

f\colonU\toV

which, when expressed in coordinates as

\psi\circf\circ\phi-1:Rm\toRn

, becomes an ordinary orthogonal projection. As an application, for each

p\inN

the corresponding fiber of

f

, denoted

Mp=f-1({p})

can be equipped with the structure of a smooth submanifold of

M

whose dimension equals the difference of the dimensions of

N

and

M

.

This theorem is a consequence of the inverse function theorem (see Inverse function theorem#Giving a manifold structure).

For example, consider

f\colonR3\toR

given by

f(x,y,z)=x4+y4+z4.

. The Jacobian matrix is
\begin{bmatrix}\partialf
\partialx

&

\partialf
\partialy

&

\partialf
\partialz

\end{bmatrix}=\begin{bmatrix}4x3&4y3&4z3\end{bmatrix}.

This has maximal rank at every point except for

(0,0,0)

. Also, the fibers

f-1(\{t\})=\left\{(a,b,c)\inR3:a4+b4+c4=t\right\}

are empty for

t<0

, and equal to a point when

t=0

. Hence, we only have a smooth submersion

f\colonR3\setminus{(0,0,0)}\toR>0,

and the subsets

Mt=\left\{(a,b,c)\inR3:a4+b4+c4=t\right\}

are two-dimensional smooth manifolds for

t>0

.

Examples

\pi\colonRm+n → Rn\subsetRm+n

Maps between spheres

A large class of examples of submersions are submersions between spheres of higher dimension, such as

f:Sn+k\toSk

whose fibers have dimension

n

. This is because the fibers (inverse images of elements

p\inSk

) are smooth manifolds of dimension

n

. Then, if we take a path

\gamma:I\toSk

and take the pullback

\begin{matrix} MI&\to&Sn+k\\ \downarrow&&\downarrowf\\ I&x → {\gamma}&Sk \end{matrix}

we get an example of a special kind of bordism, called a framed bordism. In fact, the framed cobordism groups
fr
\Omega
n
are intimately related to the stable homotopy groups.

Families of algebraic varieties

\pi:ak{X}\toS

whose fibers are smooth algebraic varieties. If we consider the underlying manifolds of these varieties, we get smooth manifolds. For example, the Weierstrass family

\pi:l{W}\toA1

of elliptic curves is a widely studied submersion because it includes many technical complexities used to demonstrate more complex theory, such as intersection homology and perverse sheaves. This family is given by

l{W}=\left\{(t,x,y)\inA1 x A2:y2=x(x-1)(x-t)\right\}

where

A1

is the affine line and

A2

is the affine plane. Since we are considering complex varieties, these are equivalently the spaces

C,C2

of the complex line and the complex plane. Note that we should actually remove the points

t=0,1

because there are singularities (since there is a double root).

Local normal form

If is a submersion at and, then there exists an open neighborhood of in, an open neighborhood of in, and local coordinates at and at such that, and the map in these local coordinates is the standard projection

f(x1,\ldots,xn,xn+1,\ldots,xm)=(x1,\ldots,xn).

It follows that the full preimage in of a regular value in under a differentiable map is either empty or a differentiable manifold of dimension, possibly disconnected. This is the content of the regular value theorem (also known as the submersion theorem). In particular, the conclusion holds for all in if the map is a submersion.

Topological manifold submersions

Submersions are also well-defined for general topological manifolds.[3] A topological manifold submersion is a continuous surjection such that for all in, for some continuous charts at and at, the map is equal to the projection map from to, where .

See also

References

Further reading

Notes and References

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