This is a basic concept in differential topology.
The map f is a submersion at a point
if its differential is a surjective linear map.
is a regular value of f if all points p in the preimage
A differentiable map f that is a submersion at each point
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.
, becomes an ordinary orthogonal projection.
can be equipped with the structure of a smooth submanifold of
The theorem is a consequence of the inverse function theorem (see Inverse function theorem#Giving a manifold structure).
The Jacobian matrix is This has maximal rank at every point except for
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.
Another large class of submersions are given by families of algebraic varieties
whose fibers are smooth algebraic varieties.
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.
Since we are considering complex varieties, these are equivalently the spaces
Note that we should actually remove the points
because there are singularities (since there is a double root).
If f: M → N is a submersion at p and f(p) = q ∈ N, 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: M → N is either empty or is a differentiable manifold of dimension dim M − dim N, possibly disconnected.
In particular, the conclusion holds for all q in N if the map f is a submersion.
Submersions are also well-defined for general topological manifolds.
[3] A topological manifold submersion is a continuous surjection f : M → N 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).