Homotopy |
In topology, two continuous function (topology) Function (mathematics) from one topological space to another are called homotopic (Greek language 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 invariant (mathematics)s 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 topology 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 s 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 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 id X and f o g is homotopic to id Y .
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: a solid disk is not homeomorphic to a single point.
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 S 1. 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 S^1 is null-homotopic precisely when it can be extended to a map of the disc D^2.
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-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
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 (mathematics), and compactification is not homotopy-invariant).
=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 function domain is the collection of all topological spaces.
An example of a 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 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 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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