Submersion (mathematics)

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In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology. The notion of a submersion is dual to the notion of an immersion.

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Definition

Let M and N be differentiable manifolds and be a differentiable map between them. The map f is a submersion at a point if its differential

is a surjective linear map. [1] In this case p is called a regular point of the map f, otherwise, p is a critical point. A point is a regular value of f if all points p in the preimage are regular points. A differentiable map f that is a submersion at each point is called a submersion. Equivalently, f is a submersion if its differential has constant rank equal to the dimension of N.

A word of warning: some authors use the term critical point to describe a point where the rank of the Jacobian matrix of f at p is not maximal. [2] Indeed, this is the more useful notion in singularity theory. If the dimension of M is greater than or equal to the dimension of N then these two notions of critical point coincide. But if the dimension of M is less than the dimension of N, 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 M). The definition given above is the more commonly used; e.g., in the formulation of Sard's theorem.

Submersion theorem

Given a submersion between smooth manifolds of dimensions and , for each there are surjective charts of around , and of around , such that restricts to a submersion which, when expressed in coordinates as , becomes an ordinary orthogonal projection. As an application, for each the corresponding fiber of , denoted can be equipped with the structure of a smooth submanifold of whose dimension is equal to the difference of the dimensions of and .

For example, consider given by The Jacobian matrix is

This has maximal rank at every point except for . Also, the fibers

are empty for , and equal to a point when . Hence we only have a smooth submersion and the subsets are two-dimensional smooth manifolds for .

Examples

Maps between spheres

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

whose fibers have dimension . This is because the fibers (inverse images of elements ) are smooth manifolds of dimension . Then, if we take a path

and take the pullback

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

Families of algebraic varieties

Another large class of submersions are given by families of algebraic varieties whose fibers are smooth algebraic varieties. If we consider the underlying manifolds of these varieties, we get smooth manifolds. For example, the Weierstauss family 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

where is the affine line and is the affine plane. Since we are considering complex varieties, these are equivalently the spaces of the complex line and the complex plane. Note that we should actually remove the points because there are singularities (since there is a double root).

Local normal form

If f: MN is a submersion at p and f(p) = qN, then there exists an open neighborhood U of p in M, an open neighborhood V of q in N, and local coordinates (x1, …, xm) at p and (x1, …, xn) at q such that f(U) = V, and the map f in these local coordinates is the standard projection

It follows that the full preimage f−1(q) in M of a regular value q in N under a differentiable map f: MN is either empty or is a differentiable manifold of dimension dim M − dim N, possibly disconnected. This is the content of the regular value theorem (also known as the submersion theorem). In particular, the conclusion holds for all q in N if the map f is a submersion.

Topological manifold submersions

Submersions are also well-defined for general topological manifolds. [3] A topological manifold submersion is a continuous surjection f : MN such that for all p in M, for some continuous charts ψ at p and φ at f(p), the map ψ−1 ∘ f ∘ φ is equal to the projection map from Rm to Rn, where m = dim(M) ≥ n = dim(N).

See also

Notes

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