Homotopy

In topology, two continuous functions from one topological space to another are called homotopic (from Ancient 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 (/həˈmɒtəpiː/,[1] hə-MO-tə-pee; /ˈhoʊmoʊˌtoʊpiː/,[2] HOH-moh-toh-pee) between the two functions.

[3] 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

An alternative notation is to say that a homotopy between two continuous functions

[4] The animation that is looped above right provides an example of a homotopy between two embeddings, f and g, of the torus into R3.

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.

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 idX and f ∘ g is homotopic to idY.

[6] Intuitively, two spaces X and Y are homotopy equivalent if they can be transformed into one another by bending, shrinking and expanding operations.

is null-homotopic precisely when it can be continuously extended to a map from the unit disk

For example, if X and Y are homotopy equivalent spaces, then: An example of an algebraic invariant of topological spaces which is not homotopy-invariant is compactly supported homology (which is, roughly speaking, the homology of the compactification, and compactification is not homotopy-invariant).

In order to define the fundamental group, one needs the notion of homotopy relative to a subspace.

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].

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.

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.

In geometric topology—for example in knot theory—the idea of isotopy is used to construct equivalence relations.

Knots K1 and K2 are considered equivalent when there is an ambient isotopy which moves K1 to K2.

On a Lorentzian manifold, certain curves are distinguished as timelike (representing something that only goes forwards, not backwards, in time, in every local frame).

, 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

We can define the action of one equivalence class on another, and so we get a group.

The homotopy category is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps.

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: Hn(f) = Hn(g) : Hn(X) → Hn(Y) for all n. Likewise, if X and Y are in addition path connected, and the homotopy between f and g is pointed, then the group homomorphisms induced by f and g on the level of homotopy groups are also the same: πn(f) = πn(g) : πn(X) → πn(Y).

Based on the concept of the homotopy, computation methods for algebraic and differential equations have been developed.

Homotopy theory can be used as a foundation for homology theory: one can represent 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

is in natural bijection with the n-th singular cohomology group

One says that the omega-spectrum of Eilenberg-MacLane spaces are representing spaces for singular cohomology with coefficients in G. Using this fact, homotopy classes between a CW complex and a multiply connected space can be calculated using cohomology as described by the Hopf–Whitney theorem.