In the
mathematical
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
field of
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, a regular homotopy refers to a special kind of
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
between
immersion
Immersion may refer to:
The arts
* "Immersion", a 2012 story by Aliette de Bodard
* ''Immersion'', a French comic book series by Léo Quievreux#Immersion, Léo Quievreux
* Immersion (album), ''Immersion'' (album), the third album by Australian gro ...
s of one
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
in another. The homotopy must be a 1-parameter family of immersions.
Similar to
homotopy class
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
es, one defines two immersions to be in the same regular homotopy class if there exists a regular homotopy between them. Regular homotopy for immersions is similar to
isotopy of embeddings: they are both restricted types of homotopies. Stated another way, two continuous functions
are homotopic if they represent points in the same path-components of the mapping space
, given the
compact-open topology In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory and ...
. The space of immersions is the subspace of
consisting of immersions, denoted by
. Two immersions
are regularly homotopic if they represent points in the same path-component of
.
Examples
Any two knots in 3-space are equivalent by regular homotopy, though not by isotopy.
![Winding Number Around Point](https://upload.wikimedia.org/wikipedia/commons/8/8f/Winding_Number_Around_Point.svg)
The Whitney–Graustein theorem classifies the regular homotopy classes of a circle into the plane; two immersions are regularly homotopic if and only if they have the same
turning number
In mathematics, the winding number or winding index 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, i.e., the curve's number of tu ...
– equivalently,
total curvature
In 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 arc length:
:\int_a^b k(s)\,ds.
The total curvature of a closed curve i ...
; equivalently, if and only if their
Gauss map
In differential geometry, the Gauss map (named after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere ''S''2. Namely, given a surface ''X'' lying in R3, the Gauss map is a continuous map ''N'': ''X'' → ''S''2 such that ' ...
s have the same degree/
winding number
In mathematics, the winding number or winding index 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, i.e., the curve's number of turn ...
.
Stephen Smale
Stephen Smale (born July 15, 1930) is an American mathematician, known for his research in topology, dynamical systems and mathematical economics. He was awarded the Fields Medal in 1966 and spent more than three decades on the mathematics facult ...
classified the regular homotopy classes of a ''k''-sphere immersed in
– they are classified by
homotopy groups
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
of
Stiefel manifolds, which is a generalization of the Gauss map, with here ''k'' partial derivatives not vanishing. More precisely, the set
of regular homotopy classes of embeddings of sphere
in
is in one-to-one correspondence with elements of group
. In case
we have
. Since
is path connected,
and
and due to Bott periodicity theorem we have
and since
then we have
. Therefore all immersions of spheres
and
in euclidean spaces of one more dimension are regular homotopic. In particular, spheres
embedded in
admit eversion if
. A corollary of his work is that there is only one regular homotopy class of a ''2''-sphere immersed in
. In particular, this means that
sphere eversions exist, i.e. one can turn the 2-sphere "inside-out".
Both of these examples consist of reducing regular homotopy to homotopy; this has subsequently been substantially generalized in the
homotopy principle (or ''h''-principle) approach.
Non-degenerate homotopy
For
locally convex, closed
space curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s, one can also define non-degenerate homotopy. Here, the 1-parameter family of immersions must be non-degenerate (i.e. the curvature may never vanish). There are 2 distinct non-degenerate homotopy classes. Further restrictions of non-vanishing torsion lead to 4 distinct equivalence classes.
References
*
*
* {{cite journal , first = Stephen , last = Smale , authorlink = Stephen Smale , title = The classification of immersions of spheres in Euclidean spaces , journal =
Annals of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
History
The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the ...
, issue = 2 , volume = 69 , date = March 1959 , pages = 327–344 , url = http://www.maths.ed.ac.uk/~aar/papers/smale4.pdf , jstor = 1970186 , doi = 10.2307/1970186
Differential topology
Algebraic topology