Homotopy

Homotopy_between_two_paths.png
In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = 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.
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Formal definitions
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
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} o f_{1} and g_{2} o g_{1} : X → Z are homotopic as well.
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. If, in addition, X and Y are pathconnected, 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).
These latter statements are the reason that algebraic topology generally can distinguish spaces only up to homotopy equivalence, to be described next.
Homotopy equivalence of spaces
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}.
The maps f and g are called homotopy equivalences in this case.
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 R^{2}  {(0,0)} is homotopy equivalent to the unit circle S^{1}. Those spaces that are homotopy equivalent to a point are called contractible.
Clearly, every homeomorphism is a homotopy equivalence, but the converse is not true: a solid disk is not homeomorphic to a single point.
Homotopyinvariant properties
Homotopy equivalence is important because in algebraic topology most concepts cannot distinguish homotopy equivalent spaces: if X and Y are homotopy equivalent, then
 if X is pathconnected, then so is Y
 if X is simply connected, then so is Y
 the homology and cohomology groups of X and Y are isomorphic
 if X and Y are pathconnected, then the fundamental groups of X and Y are isomorphic, and so are the higher homotopy groups
Homotopy category and homotopy invariants
More abstractly, one can appeal to category theory concepts. One can define the homotopy category, whose objects are topological spaces, and whose morphisms are homotopy classes of continuous maps. Two topological spaces X and Y are isomorphic in this category if and only if they are homotopyequivalent.
A homotopy invariant is any function on spaces, (or on mappings), that respects the relation of homotopy equivalence (resp. homotopy); such invariants are constitutive of homotopy theory. Of course one could have foundational objection to a function whose domain is the collection of all topological spaces.
A typical homotopy invariant is the fundamental group of a space, already mentioned earlier.
In practice homotopy theory is carried out by working with CW complexes, for technical convenience; or in some other reasonable category.
Relative homotopy
Especially 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) for all k∈K and t∈[0,1].
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 R^{2} 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 K_{1} and K_{2} in threedimensional 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 threedimensional space, and ending at a homeomorphism h such that h moves K_{1} to K_{2}.de:Homotopie fr:Homotopie ja:ホモトピー