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. 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: H_n(f) = H_n(g) : H_n(X) → H_n(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: 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 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

• if X is 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

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.