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In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, a branch of mathematics, two
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s from one
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, ; , ) between the two functions. A notable use of homotopy is the definition of
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, homoto ...
and cohomotopy groups, important
invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iterat ...
s in
algebraic topology Algebraic topology is a branch of mathematics that 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 u ...
. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with
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: :A subsp ...
s,
CW complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...
es, or spectra.


Formal definition

Formally, a homotopy between two
continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...
s ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the
product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...
of the space ''X'' with the
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. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
, 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
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''. We can also think of the second parameter as a "slider control" that allows us to smoothly transition from ''f'' to ''g'' as the slider moves from 0 to 1, and vice versa. An alternative notation is to say that a homotopy between two continuous functions f, g: X \to Y is a family of continuous functions h_t: X \to Y for t \in ,1/math> such that h_0 = f and h_1 = g, and the
map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Althou ...
(x, t) \mapsto h_t(x) is continuous from X \times ,1/math> to Y. The two versions coincide by setting h_t(x) = H(x,t). It is not sufficient to require each map h_t(x) to be continuous. The animation that is looped above right provides an example of a homotopy between two
embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is gi ...
s, ''f'' and ''g'', of the torus into . ''X'' is the torus, ''Y'' is , ''f'' is some continuous function from the torus to ''R''3 that takes the torus to the embedded surface-of-a-doughnut shape with which the animation starts; ''g'' is some continuous function that takes the torus to the embedded surface-of-a-coffee-mug shape. The animation shows the image of ''h''''t''(''x'') as a function of the parameter ''t'', where ''t'' varies with time from 0 to 1 over each cycle of the animation loop. It pauses, then shows the image as ''t'' varies back from 1 to 0, pauses, and repeats this cycle.


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 In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
on the set of all continuous functions from ''X'' to ''Y''. This homotopy relation is compatible with
function composition In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
in the following sense: if are homotopic, and are homotopic, then their compositions and are also homotopic.


Examples

* If f, g: \R \to \R^2 are given by f(x) := \left(x, x^3\right) and g(x) = \left(x, e^x\right), then the map H: \mathbb \times
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
\to \mathbb^2 given by H(x, t) = \left(x, (1 - t)x^3 + te^x\right) is a homotopy between them. * More generally, if C \subseteq \mathbb^n is a
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polyto ...
subset of
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
and f, g:
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
\to C are paths with the same endpoints, then there is a linear homotopy (or straight-line homotopy) given by *: \begin H:
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
\times
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
&\longrightarrow C \\ (s, t) &\longmapsto (1 - t)f(s) + tg(s). \end * Let \operatorname_:B^n\to B^n be the
identity function Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unch ...
on the unit ''n''- disk; i.e. the set B^n := \left\. Let c_: B^n \to B^n be the
constant function In mathematics, a constant function is a function whose (output) value is the same for every input value. For example, the function is a constant function because the value of is 4 regardless of the input value (see image). Basic properties ...
c_\vec(x) := \vec which sends every point to the
origin Origin(s) or The Origin may refer to: Arts, entertainment, and media Comics and manga * ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002 * ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...
. Then the following is a homotopy between them: *: \begin H: B^n \times
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
&\longrightarrow B^n \\ (x, t) &\longmapsto (1 - t)x. \end


Homotopy equivalence

Given two topological spaces ''X'' and ''Y'', a homotopy equivalence between ''X'' and ''Y'' is a pair of continuous
map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Althou ...
s and , such that is homotopic to the
identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unch ...
id''X'' and is homotopic to id''Y''. If such a pair exists, then ''X'' and ''Y'' are said to be homotopy equivalent, or of the same homotopy type. Intuitively, two spaces ''X'' and ''Y'' are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations. Spaces that are homotopy-equivalent to a point are called
contractible In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within th ...
.


Homotopy equivalence vs. homeomorphism

A
homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...
is a special case of a homotopy equivalence, in which is equal to the identity map id''X'' (not only homotopic to it), and is equal to id''Y''. Therefore, if X and Y are homeomorphic then they are homotopy-equivalent, but the opposite is not true. Some examples: * A solid disk is homotopy-equivalent to a single point, since you can deform the disk along radial lines continuously to a single point. However, they are not homeomorphic, since there is no
bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
between them (since one is an infinite set, while the other is finite). * The
Möbius strip In mathematics, a Möbius strip, Möbius band, or Möbius loop is a surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Benedict Listing and Aug ...
and an untwisted (closed) strip are homotopy equivalent, since you can deform both strips continuously to a circle. But they are not homeomorphic.


Examples

* The first example of a homotopy equivalence is \mathbb^n with a point, denoted \mathbb^n \simeq \. The part that needs to be checked is the existence of a homotopy H: I \times \mathbb^n \to \mathbb^n between \operatorname_ and p_0, the projection of \mathbb^n onto the origin. This can be described as H(t,\cdot) = t\cdot p_0 + (1-t)\cdot\operatorname_. * There is a homotopy equivalence between S^1 (the
1-sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
) and \mathbb^2-\. ** More generally, \mathbb^n-\ \simeq S^. * Any
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and ...
\pi: E \to B with fibers F_b homotopy equivalent to a point has homotopy equivalent total and base spaces. This generalizes the previous two examples since \pi:\mathbb^n - \ \to S^is a fiber bundle with fiber \mathbb_. * Every
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every ...
is a fiber bundle with a fiber homotopy equivalent to a point. * \mathbb^n - \mathbb^k \simeq S^ for any 0 \le k < n, by writing \mathbb^n - \mathbb^k as the total space of the fiber bundle \mathbb^k \times (\mathbb^-\)\to (\mathbb^-\), then applying the homotopy equivalences above. * If a subcomplex A of a
CW complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...
X is contractible, then the quotient space X/A is homotopy equivalent to X. * A deformation retraction is a homotopy equivalence.


Null-homotopy

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 ''f'' from the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
''S''1 to any space ''X'' is null-homotopic precisely when it can be continuously extended to a map from the
unit disk In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose ...
''D''2 to ''X'' that agrees with ''f'' on the boundary. 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.


Invariance

Homotopy equivalence is important because in
algebraic topology Algebraic topology is a branch of mathematics that 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 u ...
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: * ''X'' is
path-connected In 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 non-empty open subsets. Connectedness is one of the principal topological properties t ...
if and only if ''Y'' is. * ''X'' is
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
if and only if ''Y'' is. * The (singular) homology and
cohomology group In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
s of ''X'' and ''Y'' are
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. * If ''X'' and ''Y'' are path-connected, then the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
s of ''X'' and ''Y'' are isomorphic, and so are the higher
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, denoted \pi_1(X), which records information about loops in a space. Intuitively, homot ...
s. (Without the path-connectedness assumption, one has π1(''X'', ''x''0) isomorphic to π1(''Y'', ''f''(''x''0)) 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, and compactification is not homotopy-invariant).


Variants


Relative homotopy

In order to define the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
, 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 In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...
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 retraction from ''X'' to ''K'' and ''f'' is the identity map, this is known as a strong
deformation retract In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deforma ...
of ''X'' to ''K''. When ''K'' is a point, the term pointed homotopy is used.


Isotopy

In case the two given continuous functions ''f'' and ''g'' from the topological space ''X'' to the topological space ''Y'' are
embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is gi ...
s, one can ask whether they can be connected 'through embeddings'. 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 an embedding. A related, but different, concept is that of
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, for example a manifold, taking a submanifold to another submanifold. For example in knot theory, ...
. Requiring that two embeddings be isotopic is a stronger requirement than that they be homotopic. For example, the map from the interval ��1, 1into the real numbers defined by ''f''(''x'') = −''x'' is ''not'' isotopic to the identity ''g''(''x'') = ''x''. 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 and ''g'' has not, which is impossible under an isotopy. However, the maps are homotopic; one homotopy from ''f'' to the identity is ''H'':  ��1, 1nbsp;×  , 1nbsp;→  ��1, 1given by ''H''(''x'', ''y'') = 2''yx'' − ''x''. Two homeomorphisms (which are special cases of embeddings) of the unit ball which agree on the boundary can be shown to be isotopic using
Alexander's trick Alexander's trick, also known as the Alexander trick, is a basic result in geometric topology, named after J. W. Alexander. Statement Two homeomorphisms of the ''n''-dimensional ball D^n which agree on the boundary sphere S^ are isotopic. More ...
. For this reason, the map of the
unit disc In mathematics, the open unit disk (or disc) around ''P'' (where ''P'' is a given point in the plane), is the set of points whose distance from ''P'' is less than 1: :D_1(P) = \.\, The closed unit disk around ''P'' is the set of points whose d ...
in R2 defined by ''f''(''x'', ''y'') = (−''x'', −''y'') is isotopic to a 180-degree
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
around the origin, and so the identity map and ''f'' are isotopic because they can be connected by rotations. In
geometric topology In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originated ...
—for example in
knot theory In the mathematical field of topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot ...
—the idea of isotopy is used to construct equivalence relations. For example, when should two knots be considered the same? We take two knots, ''K''1 and ''K''2, in three-
dimension In physics and mathematics, the dimension of a mathematical 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 (1D) because only one coordin ...
al space. A knot is an
embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is gi ...
of a one-dimensional space, the "loop of string" (or the circle), into this space, and this embedding gives a homeomorphism between the circle and its image in the embedding space. The intuitive idea behind the notion of knot equivalence is that one can ''deform'' one embedding to another through a path of embeddings: a continuous function starting at ''t'' = 0 giving the ''K''1 embedding, ending at ''t'' = 1 giving the ''K''2 embedding, with all intermediate values corresponding to embeddings. This corresponds to the definition of isotopy. An
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, for example a manifold, taking a submanifold to another submanifold. For example in knot theory, ...
, studied in this context, is an isotopy of the larger space, considered in light of its action on the embedded submanifold. Knots ''K''1 and ''K''2 are considered equivalent when there is an ambient isotopy which moves ''K''1 to ''K''2. This is the appropriate definition in the topological category. Similar language is used for the equivalent concept in contexts where one has a stronger notion of equivalence. For example, a path between two smooth embeddings is a smooth isotopy.


Timelike homotopy

On a
Lorentzian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
, certain curves are distinguished as
timelike In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
(representing something that only goes forwards, not backwards, in time, in every local frame). A timelike homotopy between two timelike curves is a homotopy such that the curve remains timelike during the continuous transformation from one curve to another. No
closed timelike curve In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime, that is "closed", returning to its starting point. This possibility was first discovered by Willem Jacob van St ...
(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 In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space ...
by timelike curves. A manifold such as the
3-sphere In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensio ...
can be
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
(by any type of curve), and yet be timelike multiply connected.


Properties


Lifting and extension properties

If we have a homotopy and a cover and we are given a map such that (''h''0 is called a
lift Lift or LIFT may refer to: Physical devices * Elevator, or lift, a device used for raising and lowering people or goods ** Paternoster lift, a type of lift using a continuous chain of cars which do not stop ** Patient lift, or Hoyer lift, mobile ...
of ''h''0), then we can lift all ''H'' to a map such that The homotopy lifting property is used to characterize
fibration The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all map ...
s. Another useful property involving homotopy is the
homotopy extension property In mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space. The homotopy extension property of cofibrations is dual ...
, 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 In mathematics, in particular homotopy theory, a continuous mapping :i: A \to X, where A and X are topological spaces, is a cofibration if it lets homotopy classes of maps ,S/math> be extended to homotopy classes of maps ,S/math> whenever a map ...
s.


Groups

Since the relation of two functions f, g\colon X\to Y being homotopic relative to a subspace is an equivalence relation, we can look at the
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es of maps between a fixed ''X'' and ''Y''. If we fix X = ,1n, the unit interval , 1 crossed with itself ''n'' times, and we take its boundary \partial( ,1n) as a subspace, then the equivalence classes form a group, denoted \pi_n(Y,y_0), where y_0 is in the image of the subspace \partial( ,1n). We can define the action of one equivalence class on another, and so we get a group. These groups are called the
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, denoted \pi_1(X), which records information about loops in a space. Intuitively, homot ...
s. In the case n = 1, it is also called the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
.


Homotopy category

The idea of homotopy can be turned into a formal category of
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cat ...
. The
homotopy category In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed b ...
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 In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...
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 In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...
s induced by ''f'' and ''g'' on the level of
homology group In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolo ...
s are the same: H''n''(''f'') = H''n''(''g'') : H''n''(''X'') → H''n''(''Y'') for all ''n''. Likewise, if ''X'' and ''Y'' are in addition
path connected In 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 non-empty open subsets. Connectedness is one of the principal topological properties t ...
, and the homotopy between ''f'' and ''g'' is pointed, then the group homomorphisms induced by ''f'' and ''g'' on the level of
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, denoted \pi_1(X), which records information about loops in a space. Intuitively, homot ...
s are also the same: π''n''(''f'') = π''n''(''g'') : π''n''(''X'') → π''n''(''Y'').


Applications

Based on the concept of the homotopy, computation methods for algebraic and
differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, a ...
have been developed. The methods for algebraic equations include the
homotopy continuation Numerical algebraic geometry is a field of computational mathematics, particularly computational algebraic geometry, which uses methods from numerical analysis to study and manipulate the solutions of systems of polynomial equations. Homotopy cont ...
method and the continuation method (see
numerical continuation Numerical continuation is a method of computing approximate solutions of a system of parameterized nonlinear equations, :F(\mathbf u,\lambda) = 0. The ''parameter'' \lambda is usually a real scalar, and the ''solution'' \mathbf u an ''n''-vector ...
). The methods for differential equations include the
homotopy analysis method The homotopy analysis method (HAM) is a semi-analytical technique to solve nonlinear ordinary/ partial differential equations. The homotopy analysis method employs the concept of the homotopy from topology to generate a convergent series solut ...
. Homotopy theory can be used as a foundation for
homology theory In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolo ...
: one can
represent Represent may refer to: * ''Represent'' (Compton's Most Wanted album) or the title song, 2000 * ''Represent'' (Fat Joe album), 1993 * ''Represent'', an album by DJ Magic Mike, 1994 * "Represent" (song), by Nas, 1994 * "Represent", a song by th ...
a cohomology functor on a space ''X'' by mappings of ''X'' into an appropriate fixed space, up to homotopy equivalence. For example, for any abelian group ''G'', and any based CW-complex ''X'', the set ,K(G,n)/math> of based homotopy classes of based maps from ''X'' to the 
Eilenberg–MacLane space In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name ...
K(G,n) is in natural bijection with the ''n''-th
singular cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...
group H^n(X,G) of the space ''X''. One says that the omega-spectrum of Eilenberg-MacLane spaces are representing spaces for singular cohomology with coefficients in ''G''.


See also

* Fiber-homotopy equivalence (relative version of a homotopy equivalence) * Homeotopy *
Homotopy type theory In mathematical logic and computer science, homotopy type theory (HoTT ) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory a ...
* Mapping class group *
Poincaré conjecture In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured by H ...
*
Regular homotopy In 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. Similar to homotopy classes, one defines two imm ...


References


Sources

* * * * {{Authority control * Theory of continuous functions Maps of manifolds