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In
topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformation ...
, a branch of
mathematics Mathematics (from Greek: ) includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition. Mathematicians seek and use patterns to formulate ...
, two
continuous function In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to suff ...
s from one
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which ''closeness'' is defined but, generally, cannot be measured by a numeric distance. More specifically, a topological space is a set of points, along with a set of ...
to another are called homotopic (from
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group *Greek language, a branch of the Indo-European language family **Proto-Greek language, the assumed last common ancestor of ...
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. 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, which records information about loops in a space. Intuitively, homotopy groups record inf ...
and
cohomotopy groups In mathematics, particularly algebraic topology, cohomotopy sets are particular contravariant functors from the category of pointed topological spaces and basepoint-preserving continuous maps to the category of sets and functions. They are dual to ...
, important invariants in
algebraic topology 250px, A torus, one of the most frequently studied objects 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 cla ...
. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with
compactly generated spaceIn topology, a compactly generated space (or k-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: ...
s, CW complexes, or Spectrum (homotopy theory), spectra.


Formal definition

Formally, a homotopy between two
continuous function In mathematics, a continuous function is a function that does not have any abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its output can be assured by restricting to suff ...
s ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times [0,1] \to Y from the product topology, product of the space ''X'' with the unit interval [0, 1] to ''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 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 [0,1] such that h_0 = f and h_1 = g, and the Map_(mathematics), map (x, t) \mapsto h_t(x) is continuous from X \times [0,1] 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 embeddings, ''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 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.


Examples

* If f,g:\R\to\R^2 are given by f(x):=(x,x^3) and g(x)=(x,e^x), then the map H:\mathbb\times [0,1]\to\mathbb^2 given by H(x,t)=(x,(1-t)x^3+te^x) is a homotopy between them. * More generally, if C\subseteq\mathbb^n is a Convex set, convex subset of Euclidean space and f,g:[0,1]\to C are Path (topology), paths with the same endpoints, then there is a linear homotopy (or straight-line homotopy) given by :\begin H:[0,1]&\times [0,1]\longrightarrow C \\ (s, t) \longmapsto (1&-t)f(s)+tg(s). \end * Let \operatorname_:B^n\to B^n be the identity function on the unit ''n''-Ball (mathematics), disk, i.e. the set B^n:=\. Let c_:B^n\to B^n be the constant function c_(x):=\vec which sends every point to the Origin (mathematics), origin. Then the following is a homotopy between them: :\begin H:B^n&\times [0,1]\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 (mathematics), maps and , such that is homotopic to the identity function, identity map 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.


Homotopy equivalence vs. homeomorphism

A homeomorphism 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 between them (one way to prove this is that the disk and the point have a different dimension, and the dimension is invariant under homeomorphism). * The Möbius strip 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 and \mathbb^2-\. * More generally, \mathbb^n-\ \simeq S^. * Any fiber bundle \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 is a fiber bundle with a fiber homotopy equivalent to a point. * For any 0 \le k < n, \mathbb^n - \mathbb^k \simeq S^ by writing \mathbb^n as \mathbb^k \times \mathbb^ and applying the homotopy equivalences above. * If a subcomplex A of a CW complex X is contractible, then the quotient space (topology), 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 ''S''1 to any space ''X'' is null-homotopic precisely when it can be continuously extended to a map from the unit disk ''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 250px, A torus, one of the most frequently studied objects 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 cla ...
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 connected space, path-connected if and only if ''Y'' is. * ''X'' is simply connected if and only if ''Y'' is. * The (singular) homology (mathematics), homology and cohomology groups of ''X'' and ''Y'' are group isomorphism, isomorphic. * If ''X'' and ''Y'' are path-connected, then the fundamental groups of ''X'' and ''Y'' are isomorphic, and so are the higher homotopy groups. (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 (mathematics), compactification, and compactification is not homotopy-invariant).


Variants


Relative homotopy

In order to define the fundamental group, 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 retraction (topology), retraction 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.


Isotopy

In case the two given continuous functions ''f'' and ''g'' from the topological space ''X'' to the topological space ''Y'' are embeddings, 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. Requiring that two embeddings be isotopic is a stronger requirement than that they be homotopic. For example, the map from the interval [−1, 1] into 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, 1] × [0, 1] → [−1, 1] given 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. For this reason, the map of the unit disc in R2 defined by ''f''(''x'', ''y'') = (−''x'', −''y'') is isotopic to a 180-degree rotation around the origin, and so the identity map and ''f'' are isotopic because they can be connected by rotations. 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, ''K''1 and ''K''2, in three-dimensional space. A knot is an embedding 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, 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, certain curves are distinguished as timelike (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 (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.


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 (mathematics), lift of ''h''0), then we can lift all ''H'' to a map such that The homotopy lifting property is used to characterize fibrations. Another useful property involving homotopy is the homotopy extension property, 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 cofibrations.


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 classes of maps between a fixed ''X'' and ''Y''. If we fix X = [0,1]^n, the unit interval [0, 1] cartesian product, crossed with itself ''n'' times, and we take its Boundary (topology), boundary \partial([0,1]^n) 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([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 groups. In the case n = 1, it is also called the fundamental group.


Homotopy category

The idea of homotopy can be turned into a formal category of category theory. 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 homomorphisms induced by ''f'' and ''g'' on the level of homology groups 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 connectedness, 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 groups are also the same: π''n''(''f'') = π''n''(''g'') : π''n''(''X'') → π''n''(''Y'').


Applications

Based on the concept of the homotopy, Numerical methods, computation methods for algebraic equations, algebraic and differential equations have been developed. The methods for algebraic equations include the homotopy continuation method and the continuation method (see numerical continuation). The methods for differential equations include the homotopy analysis method.


See also

*Fiber-homotopy equivalence (relative version of a homotopy equivalence) *Homeotopy *Homotopy type theory *Mapping class group *Poincaré conjecture *Regular homotopy


References


Sources

* * * * {{Authority control Homotopy theory, Continuous mappings Maps of manifolds