An illustration of a homotopy between the two bold paths 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.
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 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.
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. If, in addition, X and Y are 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).
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 idX and f o g is homotopic to idY.
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 R2 - {(0,0)} is homotopy equivalent to the unit circle S1. 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.
Homotopy-invariant properties Homotopy equivalence is important because in algebraic topology most concepts cannot distinguish homotopy equivalent spaces: if X and Y are homotopy equivalent, then
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 homotopy-equivalent.
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 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.
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.
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