Inmathematics, animmersion is adifferentiable function betweendifferentiable manifolds whosedifferential pushforward is everywhereinjective.^[1] Explicitly,f :M →N is an immersion if

${\displaystyle D_{p}f:T_{p}M\to T_{f(p)}N}$

is an injective function at every pointp ofM (whereT_pX denotes thetangent space of a manifoldX at a pointp inX andD_p f is the derivative (pushforward) of the mapf at pointp). Equivalently,f is an immersion if its derivative has constantrank equal to the dimension ofM:^[2]

${\displaystyle \mathrm{rank}{D}_{p}f=\dim M.}$

The functionf itself need not be injective, only its derivative must be.

A related concept is that of anembedding. A smooth embedding is an injective immersionf :M →N that is also atopological embedding, so thatM isdiffeomorphic to its image inN. An immersion is precisely alocal embedding – that is, for any pointx ∈M there is aneighbourhood,U ⊆M, ofx such thatf :U →N 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]

IfM iscompact, an injective immersion is an embedding, but ifM is not compact then injective immersions need not be embeddings; compare to continuous bijections versushomeomorphisms.

Aregular homotopy between two immersionsf andg from amanifoldM to a manifoldN is defined to be a differentiable functionH :M × [0,1] →N such that for allt in[0, 1] the functionH_t :M →N defined byH_t(x) =H(x,t) for allx ∈M is an immersion, withH₀ =f,H₁ =g. A regular homotopy is thus ahomotopy through immersions.

Hassler Whitney initiated the systematic study of immersions and regular homotopies in the 1940s, proving that for2m <n + 1 every mapf :M ^m →N ⁿ of anm-dimensional manifold to ann-dimensional manifold ishomotopic to an immersion, and in fact to anembedding for2m <n; these are theWhitney immersion theorem andWhitney embedding theorem.

Stephen Smale expressed the regular homotopy classes of immersions⁠${\displaystyle f:M^m\to {\mathbb{R}}^n}$⁠ as thehomotopy groups of a certainStiefel manifold. Thesphere eversion was a particularly striking consequence.

Morris Hirsch generalized Smale's expression to ahomotopy theory description of the regular homotopy classes of immersions of anym-dimensional manifoldM ^m in anyn-dimensional manifoldN ⁿ.

The Hirsch-Smale classification of immersions was generalized byMikhail Gromov.

The primary obstruction to the existence of an immersion⁠${\displaystyle i:M^m\to {\mathbb{R}}^n}$⁠ is thestable normal bundle ofM, as detected by itscharacteristic classes, notably itsStiefel–Whitney classes. That is, since⁠${\displaystyle {\mathbb{R}}^n}$⁠ isparallelizable, the pullback of its tangent bundle toM is trivial; since this pullback is the direct sum of the (intrinsically defined) tangent bundle onM,TM, which has dimensionm, and of the normal bundleν of the immersioni, which has dimensionn −m, for there to be acodimensionk immersion ofM, there must be a vector bundle of dimensionk,ξ ^k, standing in for the normal bundleν, such that⁠${\displaystyle TM\oplus {\zeta }^k}$⁠ is trivial. Conversely, given such a bundle, an immersion ofM 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 dimensionk, it cannot come from an (unstable) normal bundle of dimension less thank. 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 spaceM 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, theMöbius strip has non-trivial tangent bundle, so it cannot immerse in codimension 0 (in⁠${\displaystyle {\mathbb{R}}^2}$⁠), though it embeds in codimension 1 (in⁠${\displaystyle {\mathbb{R}}^3}$⁠).

William S. Massey (1960) showed that these characteristic classes (the Stiefel–Whitney classes of the stable normal bundle) vanish above degreen −α(n), whereα(n) is the number of "1" digits whenn is written in binary; this bound is sharp, as realized byreal projective space. This gave evidence to theimmersion conjecture, namely that everyn-manifold could be immersed in codimensionn −α(n), i.e., in⁠${\displaystyle {\mathbb{R}}^{2n-\alpha (n)}.}$⁠ This conjecture was proven byRalph Cohen (1985).

Codimension 0 immersions are equivalentlyrelative dimension 0submersions, and are better thought of as submersions. A codimension 0 immersion of aclosed manifold is precisely acovering map, i.e., afiber bundle with 0-dimensional (discrete) fiber. ByEhresmann's theorem and Phillips' theorem on submersions, aproper 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 offundamental class and cover spaces. For instance, there is no codimension 0 immersion⁠${\displaystyle {\mathbb{S}}^1\to {\mathbb{R}}^1,}$⁠ 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 byinvariance of domain. Similarly, although⁠${\displaystyle {\mathbb{S}}^3}$⁠ and the 3-torus⁠${\displaystyle {\mathbb{T}}^3}$⁠ are both parallelizable, there is no immersion⁠${\displaystyle {\mathbb{T}}^3\to {\mathbb{S}}^3}$⁠ – 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 codimensionk immersion of a manifold corresponds to a codimension 0 immersion of ak-dimensional vector bundle, which is anopen manifold if the codimension is greater than 0, but to a closed manifold in codimension 0 (if the original manifold is closed).

Ak-tuple point (double, triple, etc.) of an immersionf :M →N is an unordered set{x₁, ...,x_k} of distinct pointsx_i ∈M with the same imagef(x_i) ∈N. IfM is anm-dimensional manifold andN is ann-dimensional manifold then for an immersionf :M →N ingeneral position the set ofk-tuple points is an(n −k(n −m))-dimensional manifold. Every embedding is an immersion without multiple points (wherek > 1). 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 insurgery theory it is necessary to decide if an immersion⁠${\displaystyle f:{\mathbb{S}}^m\to N^{2m}}$⁠ of anm-sphere in a2m-dimensional manifold is regular homotopic to an embedding, in which case it can be killed by surgery.Wall associated tof an invariantμ(f ) in a quotient of thefundamental group ring⁠${\displaystyle \mathbb{Z}[{\pi }_1(N)]}$⁠ which counts the double points off in theuniversal cover ofN. Form > 2,f is regular homotopic to an embedding if and only ifμ(f ) = 0 by theWhitney 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 byAndré 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 inknot theory. It is studied categorically via the "calculus of functors" byThomas GoodwillieArchived 2009-11-28 at theWayback Machine,John Klein, andMichael S. Weiss.

• A mathematicalrose withk petals is an immersion of the circle in the plane with a singlek-tuple point;k can be any odd number, but if even must be a multiple of 4, so the figure 8, withk = 2, is not a rose.
• TheKlein bottle, and all other non-orientable closed surfaces, can be immersed in 3-space but not embedded.
• By theWhitney–Graustein theorem, the regular homotopy classes of immersions of the circle in the plane are classified by thewinding number, which is also the number of double points counted algebraically (i.e. with signs).
• Thesphere can be turned inside out: the standard embedding⁠${\displaystyle f_0:{\mathbb{S}}^2\to {\mathbb{R}}^3}$⁠ is related to${\displaystyle f_1=-f_0:{\mathbb{S}}^2\to {\mathbb{R}}^3}$ by a regular homotopy of immersions⁠${\displaystyle f_t:{\mathbb{S}}^2\to {\mathbb{R}}^3.}$⁠
• Boy's surface is an immersion of thereal projective plane in 3-space; thus also a 2-to-1 immersion of the sphere.
• TheMorin surface is an immersion of the sphere; both it and Boy's surface arise as midway models in sphere eversion.

• 
  Boy's surface
• 
  TheMorin surface

Immersed plane curves have a well-definedturning number, which can be defined as thetotal curvature divided by 2π. This is invariant under regular homotopy, by theWhitney–Graustein theorem – topologically, it is the degree of theGauss map, or equivalently thewinding number of the unit tangent (which does not vanish) about the origin. Further, this is acomplete 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 yieldknot diagrams, which are of central interest inknot theory. While immersed plane curves, up to regular homotopy, are determined by their turning number, knots have a very rich and complex structure.

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 ofknot 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 inKoschorke's example,^[6] which is an immersed surface (formed from 3 Möbius bands, with atriple point) that does not lift to a knotted surface, but it has a double cover that does lift. A detailed analysis is given inCarter & Saito (1998a), while a more recent survey is given inCarter, Kamada & Saito (2004).

A far-reaching generalization of immersion theory is thehomotopy principle:one may consider the immersion condition (the rank of the derivative is alwaysk) as apartial 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.

• Immersed submanifold
• Isometric immersion
• Submersion

• Immersion at the Manifold Atlas
• Immersion of a manifold at the Encyclopedia of Mathematics

• Differential geometry
• Differential topology
• Maps of manifolds
• Smooth functions

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