# User Contributed Dictionary

### Noun

# Extensive Definition

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. An outstanding use of homotopy
is the definition of homotopy
groups and cohomotopy
groups, important invariants
in algebraic
topology.

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.

## Formal definition

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).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.

### 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. 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.## 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 idX and f o g is homotopic to idY.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.

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 - is homotopy
equivalent to the unit circle
S1. Those spaces that are homotopy equivalent to a point are called
contractible.

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, 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.

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. 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. 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).

## 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: Hn(f) = Hn(g) : Hn(X) → Hn(Y) for
all n. 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: πn(f) = πn(g) :
πn(X) → πn(Y).

## 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 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. A timelike homotopy between two timelike curves is a homotopy such that each intermediate curve is 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. A manifold such as the 3-sphere can be simply connected (by any type of curve), and yet be multiply timelike connected.## 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. It is useful when dealing with cofibrations.## 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'. 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. 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. For example, when should two knots be
considered the same? We take two knots, K1 and K2, in
three-dimensional
space. 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.

## See also

isotopy in German: Homotopie

isotopy in Spanish: Homotopía

isotopy in French: Homotopie

isotopy in Hebrew: הומוטופיה

isotopy in Italian: Omotopia

isotopy in Dutch: Homotopie-equivalentie

isotopy in Japanese: ホモトピー

isotopy in Polish: Homotopia

isotopy in Portuguese: Homotopia

isotopy in Russian: Гомотопия

isotopy in Serbian: Хомотопија

isotopy in Finnish: Homotopia

isotopy in Vietnamese: Đồng
luân