Английская Википедия:Immersion (mathematics)
Шаблон:Short description Шаблон:For
In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential pushforward is everywhere injective.[1] Explicitly, Шаблон:Math is an immersion if
- <math>D_pf : T_p M \to T_{f(p)}N\,</math>
is an injective function at every point Шаблон:Mvar of Шаблон:Mvar (where Шаблон:Mvar denotes the tangent space of a manifold Шаблон:Mvar at a point Шаблон:Mvar in Шаблон:Mvar). Equivalently, Шаблон:Mvar is an immersion if its derivative has constant rank equal to the dimension of Шаблон:Mvar:[2]
- <math>\operatorname{rank}\,D_p f = \dim M.</math>
The function Шаблон:Mvar itself need not be injective, only its derivative must be.
A related concept is that of an embedding. A smooth embedding is an injective immersion Шаблон:Math that is also a topological embedding, so that Шаблон:Mvar is diffeomorphic to its image in Шаблон:Mvar. An immersion is precisely a local embedding – that is, for any point Шаблон:Math there is a neighbourhood, Шаблон:Math, of Шаблон:Mvar such that Шаблон:Math is an embedding, and conversely a local embedding is an immersion.[3] For infinite dimensional manifolds, this is sometimes taken to be the definition of an immersion.[4]
If Шаблон:Mvar is compact, an injective immersion is an embedding, but if Шаблон:Mvar is not compact then injective immersions need not be embeddings; compare to continuous bijections versus homeomorphisms.
Regular homotopy
A regular homotopy between two immersions Шаблон:Mvar and Шаблон:Mvar from a manifold Шаблон:Mvar to a manifold Шаблон:Mvar is defined to be a differentiable function Шаблон:Math such that for all Шаблон:Mvar in Шаблон:Math the function Шаблон:Math defined by Шаблон:Math for all Шаблон:Math is an immersion, with Шаблон:Math, Шаблон:Math. A regular homotopy is thus a homotopy through immersions.
Classification
Hassler Whitney initiated the systematic study of immersions and regular homotopies in the 1940s, proving that for Шаблон:Math every map Шаблон:Math of an Шаблон:Mvar-dimensional manifold to an Шаблон:Mvar-dimensional manifold is homotopic to an immersion, and in fact to an embedding for Шаблон:Math; these are the Whitney immersion theorem and Whitney embedding theorem.
Stephen Smale expressed the regular homotopy classes of immersions Шаблон:Tmath as the homotopy groups of a certain Stiefel manifold. The sphere eversion was a particularly striking consequence.
Morris Hirsch generalized Smale's expression to a homotopy theory description of the regular homotopy classes of immersions of any Шаблон:Mvar-dimensional manifold Шаблон:Mvar in any Шаблон:Mvar-dimensional manifold Шаблон:Mvar.
The Hirsch-Smale classification of immersions was generalized by Mikhail Gromov.
Existence
The primary obstruction to the existence of an immersion Шаблон:Tmath is the stable normal bundle of Шаблон:Mvar, as detected by its characteristic classes, notably its Stiefel–Whitney classes. That is, since Шаблон:Tmath is parallelizable, the pullback of its tangent bundle to Шаблон:Mvar is trivial; since this pullback is the direct sum of the (intrinsically defined) tangent bundle on Шаблон:Mvar, Шаблон:Mvar, which has dimension Шаблон:Mvar, and of the normal bundle Шаблон:Mvar of the immersion Шаблон:Mvar, which has dimension Шаблон:Math, for there to be a codimension Шаблон:Mvar immersion of Шаблон:Mvar, there must be a vector bundle of dimension Шаблон:Mvar, Шаблон:Mvar, standing in for the normal bundle Шаблон:Mvar, such that Шаблон:Tmath is trivial. Conversely, given such a bundle, an immersion of Шаблон:Mvar with this normal bundle is equivalent to a codimension 0 immersion of the total space of this bundle, which is an open manifold.
The stable normal bundle is the class of normal bundles plus trivial bundles, and thus if the stable normal bundle has cohomological dimension Шаблон:Mvar, it cannot come from an (unstable) normal bundle of dimension less than Шаблон:Mvar. Thus, the cohomology dimension of the stable normal bundle, as detected by its highest non-vanishing characteristic class, is an obstruction to immersions.
Since characteristic classes multiply under direct sum of vector bundles, this obstruction can be stated intrinsically in terms of the space Шаблон:Mvar and its tangent bundle and cohomology algebra. This obstruction was stated (in terms of the tangent bundle, not stable normal bundle) by Whitney.
For example, the Möbius strip has non-trivial tangent bundle, so it cannot immerse in codimension 0 (in Шаблон:Tmath), though it embeds in codimension 1 (in Шаблон:Tmath).
Шаблон:Harvs showed that these characteristic classes (the Stiefel–Whitney classes of the stable normal bundle) vanish above degree Шаблон:Math, where Шаблон:Math is the number of "1" digits when Шаблон:Mvar is written in binary; this bound is sharp, as realized by real projective space. This gave evidence to the immersion conjecture, namely that every Шаблон:Mvar-manifold could be immersed in codimension Шаблон:Math, i.e., in Шаблон:Tmath This conjecture was proven by Шаблон:Harvs.
Codimension 0
Codimension 0 immersions are equivalently relative dimension 0 submersions, and are better thought of as submersions. A codimension 0 immersion of a closed manifold is precisely a covering map, i.e., a fiber bundle with 0-dimensional (discrete) fiber. By Ehresmann's theorem and Phillips' theorem on submersions, a proper submersion of manifolds is a fiber bundle, hence codimension/relative dimension 0 immersions/submersions behave like submersions.
Further, codimension 0 immersions do not behave like other immersions, which are largely determined by the stable normal bundle: in codimension 0 one has issues of fundamental class and cover spaces. For instance, there is no codimension 0 immersion Шаблон:Tmath despite the circle being parallelizable, which can be proven because the line has no fundamental class, so one does not get the required map on top cohomology. Alternatively, this is by invariance of domain. Similarly, although Шаблон:Tmath and the 3-torus Шаблон:Tmath are both parallelizable, there is no immersion Шаблон:Tmath – any such cover would have to be ramified at some points, since the sphere is simply connected.
Another way of understanding this is that a codimension Шаблон:Mvar immersion of a manifold corresponds to a codimension 0 immersion of a Шаблон:Mvar-dimensional vector bundle, which is an open manifold if the codimension is greater than 0, but to a closed manifold in codimension 0 (if the original manifold is closed).
Multiple points
A Шаблон:Mvar-tuple point (double, triple, etc.) of an immersion Шаблон:Math is an unordered set Шаблон:Math of distinct points Шаблон:Math with the same image Шаблон:Math. If Шаблон:Mvar is an Шаблон:Mvar-dimensional manifold and Шаблон:Mvar is an n-dimensional manifold then for an immersion Шаблон:Math in general position the set of Шаблон:Mvar-tuple points is an Шаблон:Math-dimensional manifold. Every embedding is an immersion without multiple points (where Шаблон:Math). Note, however, that the converse is false: there are injective immersions that are not embeddings.
The nature of the multiple points classifies immersions; for example, immersions of a circle in the plane are classified up to regular homotopy by the number of double points.
At a key point in surgery theory it is necessary to decide if an immersion Шаблон:Tmath of an Шаблон:Mvar-sphere in a Шаблон:Math-dimensional manifold is regular homotopic to an embedding, in which case it can be killed by surgery. Wall associated to Шаблон:Mvar an invariant Шаблон:Math in a quotient of the fundamental group ring Шаблон:Tmath which counts the double points of Шаблон:Mvar in the universal cover of Шаблон:Mvar. For Шаблон:Math, Шаблон:Mvar is regular homotopic to an embedding if and only if Шаблон:Math by the Whitney trick.
One can study embeddings as "immersions without multiple points", since immersions are easier to classify. Thus, one can start from immersions and try to eliminate multiple points, seeing if one can do this without introducing other singularities – studying "multiple disjunctions". This was first done by André Haefliger, and this approach is fruitful in codimension 3 or more – from the point of view of surgery theory, this is "high (co)dimension", unlike codimension 2 which is the knotting dimension, as in knot theory. It is studied categorically via the "calculus of functors" by Thomas Goodwillie, John Klein, and Michael S. Weiss.
Examples and properties
- A mathematical rose with Шаблон:Mvar petals is an immersion of the circle in the plane with a single Шаблон:Mvar-tuple point; Шаблон:Mvar can be any odd number, but if even must be a multiple of 4, so the figure 8, with Шаблон:Math, is not a rose.
- The Klein bottle, and all other non-orientable closed surfaces, can be immersed in 3-space but not embedded.
- By the Whitney–Graustein theorem, the regular homotopy classes of immersions of the circle in the plane are classified by the winding number, which is also the number of double points counted algebraically (i.e. with signs).
- The sphere can be turned inside out: the standard embedding Шаблон:Tmath is related to <math>f_1 = -f_0 : \mathbb S^2 \to \R^3</math> by a regular homotopy of immersions Шаблон:Tmath
- Boy's surface is an immersion of the real projective plane in 3-space; thus also a 2-to-1 immersion of the sphere.
- The Morin surface is an immersion of the sphere; both it and Boy's surface arise as midway models in sphere eversion.
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The Morin surface
Immersed plane curves
Immersed plane curves have a well-defined turning number, which can be defined as the total curvature divided by 2Шаблон:Pi. This is invariant under regular homotopy, by the Whitney–Graustein theorem – topologically, it is the degree of the Gauss map, or equivalently the winding number of the unit tangent (which does not vanish) about the origin. Further, this is a complete set of invariants – any two plane curves with the same turning number are regular homotopic.
Every immersed plane curve lifts to an embedded space curve via separating the intersection points, which is not true in higher dimensions. With added data (which strand is on top), immersed plane curves yield knot diagrams, which are of central interest in knot theory. While immersed plane curves, up to regular homotopy, are determined by their turning number, knots have a very rich and complex structure.
Immersed surfaces in 3-space
The study of immersed surfaces in 3-space is closely connected with the study of knotted (embedded) surfaces in 4-space, by analogy with the theory of knot diagrams (immersed plane curves (2-space) as projections of knotted curves in 3-space): given a knotted surface in 4-space, one can project it to an immersed surface in 3-space, and conversely, given an immersed surface in 3-space, one may ask if it lifts to 4-space – is it the projection of a knotted surface in 4-space? This allows one to relate questions about these objects.
A basic result, in contrast to the case of plane curves, is that not every immersed surface lifts to a knotted surface.[5] In some cases the obstruction is 2-torsion, such as in Koschorke's example,[6] which is an immersed surface (formed from 3 Möbius bands, with a triple point) that does not lift to a knotted surface, but it has a double cover that does lift. A detailed analysis is given in Шаблон:Harvtxt, while a more recent survey is given in Шаблон:Harvtxt.
Generalizations
Шаблон:Main A far-reaching generalization of immersion theory is the homotopy principle: one may consider the immersion condition (the rank of the derivative is always Шаблон:Mvar) as a partial differential relation (PDR), as it can be stated in terms of the partial derivatives of the function. Then Smale–Hirsch immersion theory is the result that this reduces to homotopy theory, and the homotopy principle gives general conditions and reasons for PDRs to reduce to homotopy theory.
See also
Notes
References
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External links
- Immersion at the Manifold Atlas
- Immersion of a manifold at the Encyclopedia of Mathematics
- ↑ This definition is given by Шаблон:Harvnb, Шаблон:Harvnb, Шаблон:Harvnb, Шаблон:Harvnb, Шаблон:Harvnb, Шаблон:Harvnb, Шаблон:Harvnb, Шаблон:Harvnb.
- ↑ This definition is given by Шаблон:Harvnb, Шаблон:Harvnb.
- ↑ This kind of definition, based on local diffeomorphisms, is given by Шаблон:Harvnb, Шаблон:Harvnb.
- ↑ This kind of infinite-dimensional definition is given by Шаблон:Harvnb.
- ↑ Шаблон:Harvnb; Шаблон:Harvnb
- ↑ Шаблон:Harvnb
