Encyclopedia > Homotopy

  Article Content


In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions.

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).

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 : XY are homotopic, and f2, g2 : YZ are homotopic, then their compositions f2 o f1 and g2 o g1 : XZ are homotopic as well.

This allows to define the homotopy category: the objects are topological spaces, and the 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 in the following sense: there exist continuous maps f : XY and g : YX 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.

Homotopy equivalence is important because in algebraic topology most concepts cannot distinguish homotopy equivalent spaces: if X and Y are homotopy equivalent, then

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 kK and t∈[0,1].


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.

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
North Lindenhurst, New York

... are 3,808 households out of which 37.3% have children under the age of 18 living with them, 60.9% are married couples living together, 12.2% have a female householder ...

This page was created in 28.3 ms