In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an
immersion is a differentiable map between
differentiable manifoldA differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
s whose derivative is everywhere injective.
Explicitly,
f :
M →
N is an immersion if
is an injective map at every point
p of
M (where the notation
represents the
tangent spaceIn mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
of
at the point
). Equivalently,
f is an immersion if it has constant
rank equal to the dimension of
M:
The map
f itself need not be injective, only its derivative.
A related concept is that of an
embeddingIn mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
. A smooth embedding is an injective immersion
f :
M →
N which is also a topological embedding, so that
M is diffeomorphic to its image in
N. An immersion is precisely a local embedding – i.e. for any point
there is a neighbourhood,
, of
x such that
is an embedding, and conversely a local embedding is an immersion.
If
M is compact, an injective immersion is an embedding, but if
M is not compact then injective immersions need not be embeddings; compare to continuous bijections versus
homeomorphismIn the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
s.
Regular homotopy
A
regular homotopyIn the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter family of immersions....
between two immersions
f and
g from a manifold
M to a manifold
N is defined to be a
differentiable function
H:
M × [0,1] →
N such for all
the function
Ht:
M →
N defined by
Ht(
x) =
H(
x,
t) for all
is an immersion, with
H0 =
f,
H1 =
g. A regular homotopy is thus a
homotopyIn topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
through immersions.
Classification
Hassler WhitneyHassler Whitney was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.-Work:...
initiated the systematic study of immersions and regular homotopies in the 1940s, proving that for
every map
of an
-dimensional manifold to an
-dimensional manifold is homotopic to an immersion, and in fact to an
embeddingIn mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
for
; these are the
Whitney immersion theorem and
Whitney embedding theorem.
Stephen Smale expressed the regular homotopy classes of immersions
as the homotopy groups of a certain
Stiefel manifoldIn mathematics, the Stiefel manifold Vk is the set of all orthonormal k-frames in Rn. That is, it is the set of ordered k-tuples of orthonormal vectors in Rn. It is named after Swiss mathematician Eduard Stiefel...
. The sphere eversion was a particularly striking consequence.
Morris HirschMorris William Hirsch is an American mathematician, formerly at the University of California, Berkeley.A native of Chicago, Illinois, Hirsch attained his doctorate from the University of Chicago in 1958, under supervision of Edwin Spanier and Stephen Smale. His thesis was entitled Immersions of...
generalized Smale's expression to a homotopy theory description of the regular homotopy classes of immersions of any
m-dimensional manifold
Mm in any
n-dimensional manifold
Nn.
The Hirsch-Smale classification of immersions was generalized by Mikhail Gromov.
Existence
The primary obstruction to the existence of an immersion
is the
stable normal bundleIn surgery theory, a branch of mathematics, the stable normal bundle of a differentiable manifold is an invariant which encodes the stable normal data. There are analogs for generalizations of manifold, notably PL-manifolds and topological manifolds...
of
M, as detected by its characteristic classes, notably its
Stiefel–Whitney classIn mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney class is an example of a \mathbb Z_2characteristic class associated to real vector bundles.-General presentation:...
es. That is, since
is parallelizable, the pullback of its tangent bundle to
M is trivial; since this pullback is the direct sum of the (intrinsically defined) tangent bundle on
M, TM, which has dimension
m, and of the normal bundle
of the immersion
i, which has dimension
for there to be a
codimensionIn mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...
k immersion of
M, there must be a vector bundle of dimension
k, 
standing in for the normal bundle
such that
is trivial. Conversely, given such a bundle, an immersion of
M 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
k, it cannot come from an (unstable) normal bundle of dimension less than
k. 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
M 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 stripThe Möbius strip or Möbius band is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface...
has non-trivial tangent bundle, so it cannot immerse in codimension 0 (in
), though it embeds in codimension 1 (in
).
In 1960,
William S. MasseyWilliam Schumacher Massey is an American mathematician, known for his work in algebraic topology. The Massey product is named for him. He worked also on the formulation of spectral sequences by means of exact couples, and wrote several textbooks, including Algebraic Topology .William Massey was...
showed that these characteristic classes (the Stiefel–Whitney classes of the stable normal bundle) vanish above degree
where
is the number of “1” digits when
n is written in binary; this bound is sharp, as realized by
real projective spaceIn mathematics, real projective space, or RPn, is the topological space of lines through 0 in Rn+1. It is a compact, smooth manifold of dimension n, and a special case of a Grassmannian.-Construction:...
. This gave evidence to the
Immersion Conjecture, namely that every
n-manifold could be immersed in codimension
i.e., in
and was proven in 1985 by Ralph Cohen .
Codimension 0
Codimension 0 immersions are equivalently
relative dimensionIn mathematics, specifically linear algebra and geometry, relative dimension is the dual notion to codimension.In linear algebra, given a quotient map V \to Q, the difference dim V − dim Q is the relative dimension; this equals the dimension of the kernel.In fiber bundles, the relative dimension of...
0
submersionsIn mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology...
, and are better thought of as submersions.
A codimension 0 immersion of a
closed manifoldIn mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary. In contexts where no boundary is possible, any compact manifold is a closed manifold....
is precisely a
covering mapIn mathematics, more specifically algebraic topology, a covering map is a continuous surjective function p from a topological space, C, to a topological space, X, such that each point in X has a neighbourhood evenly covered by p...
, i.e., a
fiber bundleIn mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
with 0-dimensional (discrete) fiber. By
Ehresmann's theoremIn mathematics, Ehresmann's fibration theorem states that a smooth mappingwhere M and N are smooth manifolds, such that#f is a surjective submersion, and#f is a proper map,...
and Phillip's theorem on submersions, a
properIn mathematics, a continuous function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.- Definition :...
submersion of manifolds is a fiber bundle, hence codimension/relative dimension 0 immersions/submersions behave like submersions.
Further, codimenson 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 classIn mathematics, the fundamental class is a homology class [M] associated to an oriented manifold M, which corresponds to "the whole manifold", and pairing with which corresponds to "integrating over the manifold"...
and cover spaces. For instance, there is no codimension 0 immersion
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 domainInvariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space Rn. It states:The theorem and its proof are due to L.E.J. Brouwer, published in 1912...
. Similarly, although
and the 3-torus
are both parallelizable, there is no immersion
– 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
k immersion of a manifold corresponds to a codimension 0 immersion of a
k-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
-tuple point of an immersion
f :
M →
N is an unordered set
of distinct points
with the same image
. If
M is an
-dimensional manifold and
N is an
-dimensional manifold then for an immersion
f :
M →
N in
general positionIn algebraic geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general case situation, as opposed to some more special or coincidental cases that are possible...
the set of
-tuple points is an
-dimensional manifold.
A
double point is a
-tuple point with
. An embedding is an immersion without double (or higher multiplicity) points, i.e. an embedding is an injective immersion.
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 theoryIn mathematics, specifically in geometric topology, surgery theory is a collection of techniques used to produce one manifold from another in a 'controlled' way, introduced by . Surgery refers to cutting out parts of the manifold and replacing it with a part of another manifold, matching up along...
it is necessary to decide if an immersion
of an
-sphere in a
-dimensional manifold is regular homotopic to an embedding, in which case it can be killed by surgery.
Wall associated to
f an invariant
in a quotient of the
fundamental groupIn mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
ring
which counts the double points of
f in the universal cover of
. For
is regular homotopic to an embedding if and only if
by the
WhitneyHassler Whitney was an American mathematician. He was one of the founders of singularity theory, and did foundational work in manifolds, embeddings, immersions, and characteristic classes.-Work:...
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 HaefligerAndré Haefliger is a Swiss mathematician who works primarily on topology.He studied mathematics in Lausanne. He received his PhD in 1958 from the University of Strasbourg under the supervision of Charles Ehresmann with "Structures feuilletées et cohomologie à valeurs dans un faisceau de...
, 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 theoryIn topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
. It is studied categorically via the "
calculus of functorsIn algebraic topology, a branch of mathematics, the calculus of functors or Goodwillie calculus is a technique for studying functors by approximating them by a sequence of simpler functors; it generalizes the sheafification of a presheaf...
" by
Thomas Goodwillie,
John Klein, and
Michael S. Weiss.
Examples and properties
- The Klein bottle
In mathematics, the Klein bottle is a non-orientable surface, informally, a surface in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a...
, and all other non-orientable closed surfaces, can be immersed in 3-space but not embedded.
- A mathematical rose
In mathematics, a rose or rhodonea curve is a sinusoid plotted in polar coordinates. Up to similarity, thesecurves can all be expressed by a polar equation of the form\!\,r=\cos.If k is an integer, the curve will be rose shaped with...
with 
petals is an immersion of the circle in the plane with a single 
-tuple point; k can be any odd number, but if even must be a multiple of 4, so the figure 8 is not a rose.
- By the Whitney–Graustein theorem the regular homotopy classes of immersions of the circle in the plane are classified by the winding number
In mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point...
which is also the number of double points counted algebraically (i.e. with signs).
- The sphere can be turned inside out: the standard embedding
is related to 
by a regular homotopy of immersions 
.
- Boy's surface
In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901...
is an immersion of the real projective planeIn mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...
in 3-space; thus also a 2-to-1 immersion of the sphere.
- The Morin surface
The Morin surface is the half-way model of the sphere eversion discovered by Bernard Morin. It features fourfold rotational symmetry.If the original sphere to be everted has its outer surface colored green and its inner surface colored red, then when the sphere is transformed through homotopy into...
is an immersion of the sphere; both it and Boy's surface arise as midway models in sphere eversion.
Immersed plane curves
Immersed plane curves have a well-defined turning number, which can be defined as the
total curvatureIn mathematical study of the differential geometry of curves, the total curvature of an immersed plane curve is the integral of curvature along a curve taken with respect to arclength:\int_a^b k\,ds....
divided by 2
π. This is invariant under regular homotopy, by the Whitney–Graustein theorem – topologically, it is the degree of the
Gauss mapIn differential geometry, the Gauss map maps a surface in Euclidean space R3 to the unit sphere S2. Namely, given a surface X lying in R3, the Gauss map is a continuous map N: X → S2 such that N is a unit vector orthogonal to X at p, namely the normal vector to X at p.The Gauss map can be defined...
, or equivalently the
winding numberIn mathematics, the winding number of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point...
of the unit tangent (which does not vanish) about the origin. Further, this is the only invariant – 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 theoryIn topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
. 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. In some cases the obstruction is 2-torsion, such as in
Koschorke's example, which is an immersed surface (formed from 3 Möbius bands, with a
triple pointIn thermodynamics, the triple point of a substance is the temperature and pressure at which the three phases of that substance coexist in thermodynamic equilibrium...
) that does not lift to a knotted surface, but it has a double cover that does lift. A detailed analysis is given in , while a more recent survey is given in .
Generalizations
A far-reaching generalization of immersion theory is the homotopy principle:
one may consider the immersion condition (the rank of the derivative is always
k) 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
- Submersion
In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective. This is a basic concept in differential topology...
- Immersed submanifold
- Isometric immersion