The two bold paths shown above are homotopic relative to their endpoints. Thin lines mark isocontours of one possible homotopy.

In topology, two continuous functions from one topological space to another are called homotopic (Greek homos = identical and topos = place) if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly An outstanding use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In Mathematics, homotopy groups are used in Algebraic topology to classify Topological spaces The base point preserving maps from an n -dimensional In Mathematics, particularly Algebraic topology, cohomotopy sets are particular contravariant functors from the category of pointed Topological In Mathematics, an invariant is something that does not change under a set of transformations The property of being an invariant is invariance. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic

In practice, there are technical difficulties in using homotopies with certain pathological spaces. Consequently most algebraic topologists work with compactly generated spaces, CW complexes, or spectra. In Topology, a compactly generated space (or k -space) is a Topological space whose topology is coherent with the family of all In Topology, a CW complex is a type of Topological space introduced by J In Algebraic topology, a branch of Mathematics, a spectrum is an object representing a Generalized cohomology theory.

Formal definition

A homotopy of a coffee cup into a doughnut (torus). In Geometry, a torus (pl tori) is a Surface of revolution generated by revolving a Circle in three dimensional space about an axis Coplanar

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 H: X × [0,1] → Y from the product of the space X with the unit interval [0,1] to Y such that, for all points x in X, H(x,0)=f(x) and H(x,1)=g(x). In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Topology and related areas of Mathematics, a product space is the Cartesian product of a family of Topological spaces equipped with a natural In Mathematics, the unit interval is the interval, that is the set of all Real numbers x such that zero is less than or equal to 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, at time 1 we have the function g. In Mathematics, Statistics, and the mathematical Sciences a parameter ( G auxiliary measure) is a quantity that defines certain characteristics

Properties

Continuous functions f and g (both from topological space X to Y) are said to be homotopic iff 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. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" This homotopy relation is compatible with function composition in the following sense: if f₁, g₁: X → Y are homotopic, and f₂, g₂: Y → Z are homotopic, then their compositions f₂ o f₁ and g₂ o g₁: X → Z are homotopic as well. In Mathematics, a composite function represents the application of one function to the results of another

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 maps f: X → Y and g: Y → X such that g o f is homotopic to the identity map id_X and f o g is homotopic to id_Y. In Mathematics and related technical fields the term map or mapping is often a Synonym for function. In Mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that

The maps f and g are called homotopy equivalences in this case. Clearly, 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. Topological equivalence redirects here see also Topological equivalence (dynamical systems.

Intuitively, 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 R² - {(0,0)} is homotopy equivalent to the unit circle S¹. In Mathematics, a unit circle is Those 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

A function f is said to be null-homotopic if it is homotopic to a constant function. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical (The homotopy from f to a constant function is then sometimes called a null-homotopy. ) For example, it is simple to show that a map from the circle S¹ is null-homotopic precisely when it can be extended to a map of the disc D². Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the

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 topology many concepts are homotopy invariant, that is, they respect the relation of homotopy equivalence. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic For example, if X and Y are homotopy equivalent spaces, then:

• if X is path-connected, then so is Y
• if X is simply connected, then so is Y
• 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. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is In Mathematics, specifically in Algebraic topology, cohomology is a general term for a Sequence of Abelian groups defined from a Cochain In 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 In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Mathematics, homotopy groups are used in Algebraic topology to classify Topological spaces The base point preserving maps from an n -dimensional Without the path-connectedness assumption, one has π₁(X, x₀) isomorphic to π₁(Y, f(x₀)) where f: X → Y is a homotopy equivalence and x₀ a given point in 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 Mathematics, a homology theory in Algebraic topology is compactly supported if in every degree n, the Relative homology group In Mathematics, compactification is the process or result of enlarging a Topological space to make it compact.

Homotopy category

The idea of homotopy can be turned into a formal category of category theory. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets 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. In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories

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. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Mathematics (especially Algebraic topology and Abstract algebra) homology (in Greek ὁμός homos "identical" is Likewise, if X and Y are in addition path-connected, 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). In Mathematics, connectedness is used to refer to various properties meaning in some sense "all one piece" In Mathematics, homotopy groups are used in Algebraic topology to classify Topological spaces The base point preserving maps from an n -dimensional

Relative homotopy

In order to define the fundamental group, one needs the notion of homotopy relative to a subspace. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. 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 retract from X to K and f is the identity map, this is known as a strong deformation retract of X to K.

Timelike homotopy

On a Lorentzian manifold, certain curves are distinguished as timelike. In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS A timelike homotopy between two timelike curves is a homotopy such that each intermediate curve is timelike. On a Lorentzian manifold, certain curves are distinguished as 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. In a Lorentzian manifold, a closed timelike curve (CTC is a Worldline of a material particle in Spacetime that is "closed" returning to its In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be A manifold such as the 3-sphere can be simply connected (by any type of curve), and yet be multiply timelike connected. In Mathematics, a 3-sphere is a higher-dimensional analogue of a Sphere. In Topology, a geometrical object or space is called simply connected (or 1-connected) if it is Path-connected and every path between two points can be

Homotopy extension property

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. In Mathematics, in the area of Algebraic topology, the homotopy extension property indicates when a Homotopy can be extended to another one so that the It is useful when dealing with cofibrations. In Mathematics, in particular Homotopy theory, a Continuous mapping i\colon A \to X where A and X

Isotopy

In case the two given continuous functions f and g from the topological space X to the topological space Y are homeomorphisms, one can ask whether they can be connected 'through homeomorphisms'. Topological equivalence redirects here see also Topological equivalence (dynamical systems. 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 really is a stronger requirement than that they be homotopic. 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. In Mathematics, the open unit disk around P (where P is a given point in the plane) is the set of points whose distance from P is A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation However, the map on the interval [−1,1] in R defined by f(x) = −x is not isotopic to the identity. Loosely speaking, 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.

In geometric topology—for example in knot theory—the idea of isotopy is used to construct equivalence relations. In Mathematics, geometric topology is the study of Manifolds and their Embeddings Low-dimensional topology, concerning questions of dimensions In Mathematics, knot theory is the area of Topology that studies mathematical knots While inspired by knots which appear in daily life in shoelaces For example, when should two knots be considered the same? We take two knots, K₁ and K₂, in three-dimensional space. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it The intuitive idea of deforming one to the other should correspond to a path of homeomorphisms: an isotopy starting with the identity homeomorphism of three-dimensional space, and ending at a homeomorphism, h, such that h moves K₁ to K₂. An ambient isotopy, studied in this context, is an isotopy of the larger space, considered in light of its action on the embedded submanifold. 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

See also

• Mapping class group
• Homeotopy
• Regular homotopy