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In topology, a branch of mathematics, two continuous functions from one topological space to another are called **homotopic** (from 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. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.^{ [1] }

- Formal definition
- Properties
- Examples
- Homotopy equivalence
- Homotopy equivalence vs. homeomorphism
- Examples 2
- Null-homotopy
- Invariance
- Variants
- Relative homotopy
- Isotopy
- Timelike homotopy
- Properties 2
- Lifting and extension properties
- Groups
- Homotopy category
- Applications
- See also
- References
- Sources

In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, 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 and for all .

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 is a family of continuous functions for such that and , and the map is continuous from to . The two versions coincide by setting . It is not sufficient to require each map to be continuous.^{ [2] }

The animation that is looped above right provides an example of a homotopy between two embeddings, *f* and *g*, of the torus into *R*^{3}. *X* is the torus, *Y* is *R*^{3}, *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.

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 *f*_{1}, *g*_{1} : *X* → *Y* are homotopic, and *f*_{2}, *g*_{2} : *Y* → *Z* are homotopic, then their compositions *f*_{2} ∘ *f*_{1} and *g*_{2} ∘ *g*_{1} : *X* → *Z* are also homotopic.

- If are given by and , then the map given by is a homotopy between them.
- More generally, if is a convex subset of Euclidean space and are paths with the same endpoints, then there is a
**linear homotopy**^{ [3] }(or**straight-line homotopy**) given by

- Let be the identity function on the unit
*n*-disk, i.e. the set . Let be the constant function which sends every point to the origin. Then the following is a homotopy between them:

Given two topological spaces *X* and *Y*, a **homotopy equivalence** between X and Y is a pair of continuous maps *f* : *X* → *Y* and *g* : *Y* → *X*, such that *g* ∘ *f* is homotopic to the identity map id_{X} and *f* ∘ *g* 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.

A homeomorphism is a special case of a homotopy equivalence, in which *g* ∘ *f* is equal to the identity map id_{X} (not only homotopic to it), and *f* ∘ *g* is equal to id_{Y}.^{ [4] }^{:0:53:00} 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 (since one is an infinite set, while the other is finite).
- 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.

- The first example of a homotopy equivalence is with a point, denoted . The part that needs to be checked is the existence of a homotopy between and , the projection of onto the origin. This can be described as .
- There is a homotopy equivalence between and .
- More generally, .
- Any fiber bundle with fibers homotopy equivalent to a point has homotopy equivalent total and base spaces. This generalizes the previous two examples since is a fiber bundle with fiber .
- Every vector bundle is a fiber bundle with a fiber homotopy equivalent to a point.
- For any , by writing as and applying the homotopy equivalences above.
- If a subcomplex of a CW complex is contractible, then the quotient space is homotopy equivalent to .
^{ [5] } - A deformation retraction is a homotopy equivalence.

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.

Homotopy equivalence is important because in 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:

*X*is path-connected if and only if*Y*is.*X*is simply connected if and only if*Y*is.- The (singular) homology and cohomology groups of
*X*and*Y*are 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 π_{1}(*X*,*x*_{0}) isomorphic to π_{1}(*Y*,*f*(*x*_{0})) where*f*:*X*→*Y*is a homotopy equivalence and*x*_{0}∈*X*.)

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).

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 *H* : *X*× [0, 1] → *Y* between *f* and *g* such that *H*(*k*, *t*) = *f*(*k*) = *g*(*k*) for all *k* ∈ *K* and *t* ∈ [0, 1]. Also, if *g* is a 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.

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.^{ [6] }

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 **R**^{2} 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**.

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.^{ [7] }

If we have a homotopy *H* : *X*× [0,1] →*Y* and a cover *p* : *Y*→*Y* and we are given a map *h*_{0} : *X*→*Y* such that *H*_{0} = *p* ○ *h*_{0} (*h*_{0} is called a lift of *h*_{0}), then we can lift all *H* to a map *H* : *X*× [0, 1] →*Y* such that *p* ○ *H* = *H*. 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.

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] crossed with itself *n* times, and we take its boundary as a subspace, then the equivalence classes form a group, denoted , where is in the image of the subspace .

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 , it is also called the fundamental group.

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 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*).

Based on the concept of the homotopy, computation methods for algebraic and differential equations have been developed. The methods for algebraic equations include the homotopy continuation method^{ [8] } and the continuation method (see numerical continuation). The methods for differential equations include the homotopy analysis method.

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

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 the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent have isomorphic fundamental groups.

In the mathematical field of topology, a **homeomorphism**, **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 the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called **homeomorphic**, and from a topological viewpoint they are the same. The word *homeomorphism* comes from the Greek words *ὅμοιος* (*homoios*) = similar or same and *μορφή* (*morphē*) = shape, form, introduced to mathematics by Henri Poincaré in 1895.

In mathematics, specifically in homology theory and algebraic topology, **cohomology** is a general term for a sequence of abelian groups associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory.

In mathematics, specifically algebraic topology, a **covering map** is a continuous function from a topological space to a topological space such that each point in has an open neighborhood **evenly covered** by . In this case, is called a **covering space** and the **base space** of the covering projection. The definition implies that every covering map is a local homeomorphism.

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 information about the basic shape, or *holes*, of a topological space.

A **CW complex** is a kind of a topological space that is particularly important in algebraic topology. It was 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 nature that allows for computation. The *C* stands for "closure-finite", and the *W* for "weak" topology. A CW complex can be defined inductively.

In algebraic topology, a branch of mathematics, **singular homology** refers to the study of a certain set of algebraic invariants of a topological space *X*, the so-called **homology groups** Intuitively, singular homology counts, for each dimension *n*, the *n*-dimensional holes of a space. Singular homology is a particular example of a homology theory, which has now grown to be a rather broad collection of theories. Of the various theories, it is perhaps one of the simpler ones to understand, being built on fairly concrete constructions.

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 functional analysis. It was introduced by Ralph Fox in 1945.

In mathematics, in the subfield of geometric topology, the **mapping class group** is an important algebraic invariant of a topological space. Briefly, the mapping class group is a certain discrete group corresponding to symmetries of the space.

In mathematics, a **path** in a topological space is a continuous function from the closed unit interval into

In mathematics, in particular homotopy theory, a continuous mapping

In mathematics, the **Teichmüller space** of a (real) topological surface , is a space that parametrizes complex structures on up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüller spaces are named after Oswald Teichmüller.

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 categories, as discussed below.

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 **deformation retraction** is a mapping that captures the idea of *continuously shrinking* a space into a subspace.

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.

In surgery theory, a branch of mathematics, the **stable normal bundle** of a differentiable manifold is an invariant which encodes the stable normal data. There are analogs for generalizations of manifold, notably PL-manifolds and topological manifolds. There is also an analogue in homotopy theory for Poincaré spaces, the **Spivak spherical fibration**, named after Michael Spivak.

In the mathematical subject of geometric group theory, the **Culler–Vogtmann Outer space** or just **Outer space** of a free group *F*_{n} is a topological space consisting of the so-called "marked metric graph structures" of volume 1 on *F*_{n}. The Outer space, denoted *X*_{n} or *CV*_{n}, comes equipped with a natural action of the group of outer automorphisms Out(*F*_{n}) of *F*_{n}. The Outer space was introduced in a 1986 paper, of Marc Culler and Karen Vogtmann and it serves as a free group analog of the Teichmüller space of a hyperbolic surface. Outer space is used to study homology and cohomology groups of Out(*F*_{n}) and to obtain information about algebraic, geometric and dynamical properties of Out(*F*_{n}), of its subgroups and individual outer automorphisms of *F*_{n}. The space *X*_{n} can also be thought of as the set of *F*_{n}-equivariant isometry types of minimal free discrete isometric actions of *F*_{n} on *F*_{n} on **R**-trees*T* such that the quotient metric graph *T*/*F*_{n} has volume 1.

In mathematics, especially in the area of topology known as algebraic topology, an **induced homomorphism** is a homomorphism derived in a canonical way from another map. For example, a continuous map from a topological space *X* to a space *Y* induces a group homomorphism from the fundamental group of *X* to the fundamental group of *Y*.

This is a glossary of properties and concepts in algebraic topology in mathematics.

In mathematics, **homotopy theory** is a systematic study of situations in which maps come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry and category theory.

- ↑ "Homotopy | mathematics".
*Encyclopedia Britannica*. Retrieved 2019-08-17. - ↑ Path homotopy and separately continuous functions
- ↑ Allen., Hatcher (2002).
*Algebraic topology*. Cambridge: Cambridge University Press. p. 185. ISBN 9780521795401. OCLC 45420394. - ↑ Albin, Pierre (2019). "History of algebraic topology".
- ↑ Allen., Hatcher (2002).
*Algebraic topology*. Cambridge: Cambridge University Press. p. 11. ISBN 9780521795401. OCLC 45420394. - ↑ Weisstein, Eric W. "Isotopy".
*MathWorld*. - ↑ Monroe, Hunter (2008-11-01). "Are Causality Violations Undesirable?".
*Foundations of Physics*.**38**(11): 1065–1069. arXiv: gr-qc/0609054 . Bibcode:2008FoPh...38.1065M. doi:10.1007/s10701-008-9254-9. ISSN 0015-9018. - ↑ Allgower, Eugene; Georg, Kurt. "Introduction to Numerical Continuation Methods" (PDF). CSU. Retrieved 22 February 2020.

- Armstrong, M.A. (1979).
*Basic Topology*. Springer. ISBN 978-0-387-90839-7. - "Homotopy",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - "Isotopy (in topology)",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Spanier, Edwin (December 1994).
*Algebraic Topology*. Springer. ISBN 978-0-387-94426-5.

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