Homotopy
Encyclopedia
In topology
, two continuous functions
from one topological space
to another are called homotopic (Greek
ὁμός (homós) = same, similar, and τόπος (tópos) = place) if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. An outstanding use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariant
s in algebraic topology
.
In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated space
s, CW complex
es, or spectra
.
Formally, a homotopy between two continuous functions f and g from a
topological space X to a topological space Y is defined to be a continuous function from the product
of the space X with the unit interval
[0,1] to Y such that, if then and
If we think of the second parameter
of H as time then H describes a continuous deformation of f into g: at time 0 we have the function f and at time 1 we have the function g.
An alternative notation is to say that a homotopy between two continuous functions is a family of continuous functions for such that and and the map is continuous from [0,1] to the space of all continuous functions The two versions coincide by setting
Being homotopic is an equivalence relation
on the set of all continuous functions from X to Y.
This homotopy relation is compatible with function composition
in the following sense: if are homotopic, and are homotopic, then their compositions and are also homotopic.
s and such that is homotopic to the identity map
idX and is homotopic to idY.
The maps f and g are called homotopy equivalences in this case. Every homeomorphism
is a homotopy equivalence, but the converse is not true: for example, a solid disk is not homeomorphic to a single point, although the disk and the point are homotopy equivalent.
Two spaces X and Y are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations. For example, a solid disk or solid ball is homotopy equivalent to a point, and } is homotopy equivalent to the unit circle
S1. Spaces that are homotopy equivalent to a point are called contractible.
A function f is said to be null-homotopic if it is homotopic to a constant function. (The homotopy from f to a constant function is then sometimes called a null-homotopy.) For example, a map from the circle
S1 is null-homotopic precisely when it can be extended to a map of the disc D2.
It follows from these definitions that a space X is contractible if and only if the identity map from X to itself—which is always a homotopy equivalence—is null-homotopic.
many concepts are homotopy invariant, that is, they respect the relation of homotopy equivalence. For example, if X and Y are homotopy equivalent spaces, then:
An example of an algebraic invariant of topological spaces which is not homotopy-invariant is compactly supported homology (which is, roughly speaking, the homology of the compactification
, and compactification is not homotopy-invariant).
, one needs the notion of homotopy relative to a subspace. These are homotopies which keep the elements of the subspace fixed. Formally: if f and g are continuous maps from X to Y and K is a subset
of X, then we say that f and g are homotopic relative to K if there exists a homotopy between f and g such that for all and Also, if g is a retract from X to K and f is the identity map, this is known as a strong deformation retract
of X to K.
When K is a point, the term pointed homotopy is used.
with itself n times, and we take our subspace to be its boundary
∂([0,1]n) then the equivalence classes form a group, denoted πn(Y,y0), where y0 is in the image of the subspace ∂([0,1]n).
We can define the action of one equivalence class on another, and so we get a group. These groups are called the homotopy group
s. In the case it is also called the fundamental group
.
. The homotopy category is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. Two topological spaces X and Y are isomorphic in this category if and only if they are homotopy-equivalent. Then a functor
on the category of topological spaces is homotopy invariant if it can be expressed as a functor on the homotopy category.
For example, homology groups are a functorial homotopy invariant: this means that if f and g from X to Y are homotopic, then the group homomorphism
s induced by f and g on the level of homology groups are the same: Hn(f) = Hn(g) : Hn(X) → Hn(Y) for all n. Likewise, if X and Y are in addition path connected
, and the homotopy between f and g is pointed, then the group homomorphisms induced by f and g on the level of homotopy group
s are also the same: πn(f) = πn(g) : πn(X) → πn(Y).
between two timelike curves is a homotopy such that each intermediate curve is timelike. No closed timelike curve
(CTC) on a Lorentzian manifold is timelike homotopic to a point (that is, null timelike homotopic); such a manifold is therefore said to be multiply connected by timelike curves. A manifold such as the 3-sphere
can be simply connected (by any type of curve), and yet be timelike multiply connected.http://dx.doi.org/10.1007/s10701-008-9254-9
of h0), then we can lift all H to a map such that The homotopy lifting property is used to characterize fibration
s.
,
which characterizes the extension of a homotopy between two functions from a subset of some set to the set itself. It is useful when dealing with cofibration
s.
s, one can ask whether they can be connected 'through homeomorphisms'. This gives rise to the concept of isotopy, which is a homotopy, H, in the notation used before, such that for each fixed t, H(x,t) gives a homeomorphism.
Requiring that two homeomorphisms be isotopic is a stronger requirement than that they be homotopic. Unit balls which agree on the boundary can be shown to be isotopic using Alexander's trick.
For example, the map of the unit disc in R2 defined by f(x,y) = (−x, −y) is equivalent to a 180-degree rotation
around the origin, and so the identity map and f are isotopic because they can be connected by rotations. However, the map on the interval [−1,1] in R defined by f(x) = −x is not isotopic to the identity. Any homotopy from f to the identity would have to exchange the endpoints, which would mean that they would have to 'pass through' each other. Moreover, f has changed the orientation of the interval, hence it cannot be isotopic to the identity. However, the maps are homotopic; one homotopy from f to the identity is H: [−1,1] × [0,1] → [−1,1] given by H(x,y) = 2yx-x.
In geometric topology
—for example in knot theory
—the idea of isotopy is used to construct equivalence relations. For example, when should two knots be considered the same? We take two knots, K1 and K2, in three-dimension
al space. A knot is an embedding
of a one-dimensional space, the "loop of string", into this space, and an embedding is simply a homeomorphism. The intuitive idea of deforming one to the other should correspond to a path of embeddings: a continuous function starting at t=0 with the K1 embedding, ending at t=1 with the K2 embedding, with all intermediate values being embeddings; this corresponds to the definition of isotopy. However, this does not distinguish knots because the knotted portion can be isotoped down to a point, leaving an unknotted circle. An ambient isotopy
, studied in this context, is an isotopy of the larger space, considered in light of its action on the embedded submanifold. Knots K1 and K2 are considered equivalent when there is an ambient isotopy which moves K1 to K2.
.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, two continuous functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
from one topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
to another are called homotopic (Greek
Greek language
Greek is an independent branch of the Indo-European family of languages. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history;...
ὁμός (homós) = same, similar, and τόπος (tópos) = place) if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. An outstanding use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariant
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
s in algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
.
In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated space
Compactly generated space
In topology, a compactly generated space is a topological space whose topology is coherent with the family of all compact subspaces. Specifically, a topological space X is compactly generated if it satisfies the following condition:Equivalently, one can replace closed with open in this definition...
s, CW complex
CW complex
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
es, or spectra
Spectrum (homotopy theory)
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. There are several different constructions of categories of spectra, any of which gives a context for the same stable homotopy theory....
.
Formal definition
topological space X to a topological space Y is defined to be a continuous function from the product
Product topology
In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...
of the space X with the unit interval
Unit interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...
[0,1] to Y such that, if then and
If we think of the second parameter
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
of H as time then H describes a continuous deformation of f into g: at time 0 we have the function f and at time 1 we have the function g.
An alternative notation is to say that a homotopy between two continuous functions is a family of continuous functions for such that and and the map is continuous from [0,1] to the space of all continuous functions The two versions coincide by setting
Properties
Continuous functions f and g are said to be homotopic if and only if there is a homotopy H taking f to g as described above.Being homotopic is an equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
on the set of all continuous functions from X to Y.
This homotopy relation is compatible with function composition
Function composition
In mathematics, function composition is the application of one function to the results of another. For instance, the functions and can be composed by computing the output of g when it has an argument of f instead of x...
in the following sense: if are homotopic, and are homotopic, then their compositions and are also homotopic.
Homotopy equivalence and null-homotopy
Given two spaces X and Y, we say they are homotopy equivalent or of the same homotopy type if there exist continuous mapMap (mathematics)
In most of mathematics and in some related technical fields, the term mapping, usually shortened to map, is either a synonym for function, or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function.In graph theory, a map is a...
s and such that is homotopic to the identity map
Identity function
In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...
idX and is homotopic to idY.
The maps f and g are called homotopy equivalences in this case. Every homeomorphism
Homeomorphism
In 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...
is a homotopy equivalence, but the converse is not true: for example, a solid disk is not homeomorphic to a single point, although the disk and the point are homotopy equivalent.
Two spaces X and Y are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations. For example, a solid disk or solid ball is homotopy equivalent to a point, and } is homotopy equivalent to the unit circle
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
S1. Spaces that are homotopy equivalent to a point are called contractible.
A function f is said to be null-homotopic if it is homotopic to a constant function. (The homotopy from f to a constant function is then sometimes called a null-homotopy.) For example, a map from the circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
S1 is null-homotopic precisely when it can be extended to a map of the disc D2.
It follows from these definitions that a space X is contractible if and only if the identity map from X to itself—which is always a homotopy equivalence—is null-homotopic.
Homotopy invariance
Homotopy equivalence is important because in algebraic topologyAlgebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
many concepts are homotopy invariant, that is, they respect the relation of homotopy equivalence. For example, if X and Y are homotopy equivalent spaces, then:
- If X is path-connectedConnected spaceIn topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
then so is Y. - If X is simply connected then so is Y.
- The (singular) homologyHomology (mathematics)In mathematics , homology is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group...
and cohomology groups of X and Y are isomorphicGroup isomorphismIn abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic...
. - If X and Y are path-connected, then the fundamental groupFundamental 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...
s of X and Y are isomorphic, and so are the higher homotopy groupHomotopy groupIn mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
s. (Without the path-connectedness assumption, one has π1(X,x0) isomorphic to π1(Y,f(x0)) where is a homotopy equivalence and
An example of an algebraic invariant of topological spaces which is not homotopy-invariant is compactly supported homology (which is, roughly speaking, the homology of the compactification
Compactification (mathematics)
In mathematics, compactification is the process or result of making a topological space compact. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".-An...
, and compactification is not homotopy-invariant).
Relative homotopy
In order to define the fundamental groupFundamental group
In 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...
, one needs the notion of homotopy relative to a subspace. These are homotopies which keep the elements of the subspace fixed. Formally: if f and g are continuous maps from X to Y and K is a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of X, then we say that f and g are homotopic relative to K if there exists a homotopy between f and g such that for all and Also, if g is a retract from X to K and f is the identity map, this is known as a strong deformation retract
Deformation retract
In topology, a branch of mathematics, a retraction , as the name suggests, "retracts" an entire space into a subspace. A deformation retraction is a map which captures the idea of continuously shrinking a space into a subspace.- Retract :...
of X to K.
When K is a point, the term pointed homotopy is used.
Homotopy groups
Since the relation of two functions being homotopic relative to a subspace is an equivalence relation, we can look at the equivalence classes of maps between a fixed X and Y. If we fix the unit interval [0,1] crossedCartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
with itself n times, and we take our subspace to be its boundary
Boundary (topology)
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S, not belonging to the interior of S. An element of the boundary...
∂([0,1]n) then the equivalence classes form a group, denoted πn(Y,y0), where y0 is in the image of the subspace ∂([0,1]n).
We can define the action of one equivalence class on another, and so we get a group. These groups are called the homotopy group
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
s. In the case it is also called the fundamental group
Fundamental group
In 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...
.
Homotopy category
The idea of homotopy can be turned into a formal category of category theoryCategory theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
. The homotopy category is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. Two topological spaces X and Y are isomorphic in this category if and only if they are homotopy-equivalent. Then a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
on the category of topological spaces is homotopy invariant if it can be expressed as a functor on the homotopy category.
For example, homology groups are a functorial homotopy invariant: this means that if f and g from X to Y are homotopic, then the group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
s induced by f and g on the level of homology groups are the same: Hn(f) = Hn(g) : Hn(X) → Hn(Y) for all n. Likewise, if X and Y are in addition path connected
Connectedness
In mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected...
, and the homotopy between f and g is pointed, then the group homomorphisms induced by f and g on the level of homotopy group
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
s are also the same: πn(f) = πn(g) : πn(X) → πn(Y).
Timelike homotopy
On a Lorentzian manifold, certain curves are distinguished as timelike. A timelike homotopyTimelike homotopy
On a Lorentzian manifold, certain curves are distinguished as timelike. A timelike homotopy between two timelike curves is a homotopy such that each intermediate curve is timelike...
between two timelike curves is a homotopy such that each intermediate curve is timelike. No closed timelike curve
Closed timelike curve
In mathematical physics, a closed timelike curve is a worldline in a Lorentzian manifold, of a material particle in spacetime that is "closed," returning to its starting point...
(CTC) on a Lorentzian manifold is timelike homotopic to a point (that is, null timelike homotopic); such a manifold is therefore said to be multiply connected by timelike curves. A manifold such as the 3-sphere
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...
can be simply connected (by any type of curve), and yet be timelike multiply connected.http://dx.doi.org/10.1007/s10701-008-9254-9
Homotopy lifting property
If we have a homotopy and a cover and we are given a map such that (h0 is called a liftLift (mathematics)
In the branch of mathematics called category theory, given a morphism f from an object X to an object Y, and a morphism g from an object Z to Y, a lift of f to Z is a morphism h from X to Z such that gh = f.A basic example in topology is lifting a path in one space to a path in a covering space...
of h0), then we can lift all H to a map such that The homotopy lifting property is used to characterize fibration
Fibration
In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...
s.
Homotopy extension property
Another useful property involving homotopy is the homotopy extension propertyHomotopy extension property
In mathematics, in the area of algebraic topology, the homotopy extension property indicates when a homotopy defined on a subspace can be extended to a homotopy defined on a larger space.-Definition:Let X\,\! be a topological space, and let A \subset X....
,
which characterizes the extension of a homotopy between two functions from a subset of some set to the set itself. It is useful when dealing with cofibration
Cofibration
In mathematics, in particular homotopy theory, a continuous mappingi\colon A \to X,where A and X are topological spaces, is a cofibration if it satisfies the homotopy extension property with respect to all spaces Y. The name is because the dual condition, the homotopy lifting property, defines...
s.
Isotopy
In case the two given continuous functions f and g from the topological space X to the topological space Y are homeomorphismHomeomorphism
In 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, one can ask whether they can be connected 'through homeomorphisms'. This gives rise to the concept of isotopy, which is a homotopy, H, in the notation used before, such that for each fixed t, H(x,t) gives a homeomorphism.
Requiring that two homeomorphisms be isotopic is a stronger requirement than that they be homotopic. Unit balls which agree on the boundary can be shown to be isotopic using Alexander's trick.
For example, the map of the unit disc in R2 defined by f(x,y) = (−x, −y) is equivalent to a 180-degree rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
around the origin, and so the identity map and f are isotopic because they can be connected by rotations. However, the map on the interval [−1,1] in R defined by f(x) = −x is not isotopic to the identity. Any homotopy from f to the identity would have to exchange the endpoints, which would mean that they would have to 'pass through' each other. Moreover, f has changed the orientation of the interval, hence it cannot be isotopic to the identity. However, the maps are homotopic; one homotopy from f to the identity is H: [−1,1] × [0,1] → [−1,1] given by H(x,y) = 2yx-x.
In geometric topology
Geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...
—for example in knot theory
Knot theory
In 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...
—the idea of isotopy is used to construct equivalence relations. For example, when should two knots be considered the same? We take two knots, K1 and K2, in three-dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
al space. A knot is an embedding
Embedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
of a one-dimensional space, the "loop of string", into this space, and an embedding is simply a homeomorphism. The intuitive idea of deforming one to the other should correspond to a path of embeddings: a continuous function starting at t=0 with the K1 embedding, ending at t=1 with the K2 embedding, with all intermediate values being embeddings; this corresponds to the definition of isotopy. However, this does not distinguish knots because the knotted portion can be isotoped down to a point, leaving an unknotted circle. An ambient isotopy
Ambient isotopy
In the mathematical subject of topology, an ambient isotopy, also called an h-isotopy, is a kind of continuous distortion of an "ambient space", a manifold, taking a submanifold to another submanifold. For example in knot theory, one considers two knots the same if one can distort one knot into the...
, studied in this context, is an isotopy of the larger space, considered in light of its action on the embedded submanifold. Knots K1 and K2 are considered equivalent when there is an ambient isotopy which moves K1 to K2.
Applications
Based on the concept of the homotopy, computation methods for algebraic and differential equations are developed. The methods for algebraic equations include the homotopy continuation method and the continuation method. The methods for differential equations include the homotopy analysis methodHomotopy analysis method
The homotopy analysis method aims to solve nonlinear ordinary differential equations and partial differential equations analytically. The method distinguishes itself from other analytical methods in the following four aspects. First, it is a series expansion method but it is independent of small...
.
See also
- Mapping class groupMapping class groupIn mathematics, in the sub-field of geometric topology, the mapping class groupis an important algebraic invariant of a topological space. Briefly, the mapping class group is a discrete group of 'symmetries' of the space.-Motivation:...
- HomeotopyHomeotopyIn algebraic topology, an area of mathematics, a homeotopy group of a topological space is a homotopy group of the group of self-homeomorphisms of that space.-Definition:...
- Regular homotopyRegular 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....
- Poincaré conjecturePoincaré conjectureIn mathematics, the Poincaré conjecture is a theorem about the characterization of the three-dimensional sphere , which is the hypersphere that bounds the unit ball in four-dimensional space...
- Homotopy analysis methodHomotopy analysis methodThe homotopy analysis method aims to solve nonlinear ordinary differential equations and partial differential equations analytically. The method distinguishes itself from other analytical methods in the following four aspects. First, it is a series expansion method but it is independent of small...