Homotopy

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 : 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.

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 : 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 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

• if X is path-connected, then so is Y
• if X is locally path-connected, 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 path-connected, then the fundamental groups of X and Y are isomorphic, and so are the higher homotopy groups[?]

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.