In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The many different ways to (continuously) map an n-dimensional sphere into a given space are collected into equivalence classes, called homotopy classes. Two mappings are homotopic if one can be continuously deformed into the other. These homotopy classes form a group, called the n-th homotopy group of the given space. Topological spaces with differing homotopy groups are never equivalent (homeomorphic), but the converse is not true. The first homotopy group is also called the fundamental group.

Homotopy groups

In the sphere Sⁿ we choose a base point a. For a space X with base point b, we define π_n(X) to be the set of homotopy classes of maps f : Sⁿ → X that map the base point a to the base point b. In particular, the equivalence clases are given by homotopies that are constant on the basepoint of the sphere. Equivalently, we can define π_n(X) to be the group of homotopy classes of maps g : [0,1]ⁿ → X from the n-cube to X that take the boundary of the n-cube to b.

For n ≥ 1, the homotopy classes form a group. To define the group operation, recall that in the fundamental group, the product f * g of two loops f and g is defined by setting (f * g)(t) = f(2t) if t is in [0,1/2] and (f * g)(t) = g(2t-1) if t is in [1/2,1]. The idea of composition in the fundamental group is that of following the first path and the second in succession, or, equivalently, setting their two domains together. The concept of composition that we want for the n-th homotopy group is the same, except that now the domains that we stick together are cubes, and we must glue them along a face. We therefore define the sum of maps f, g : [0,1]ⁿ → X by the formula (f + g)(t₁, t₂, … t_n) = f(2t₁, t₂, … t_n) for t₁ in [0,1/2] and (f + g)(t₁, t₂, … t_n) = g(2t₁-1, t₂, … t_n) for t₁ in [1/2,1]. For the corresponding definition in terms of spheres, define the sum f + g of maps f, g : Sⁿ → X to be k composed with h, where k is the map from Sⁿ to the wedge sum of two n-spheres that collapses the equator and h is the map from the wedge sum of two n-spheres to X that is defined to be f on the first sphere and g on the second.

If n ≥ 2, then π_n is abelian. (For a proof of this, note that in two dimensions or greater, two homotopies can be "rotated" around each other.)

The long exact sequence of a fibration

Let p : E → B be a basepoint-preserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Then there is a long exact sequence of homotopy groups

… → π_n(F) → π_n(E) → π_n(B) → π_n−1(F) → … → π₀(E) → π₀(B) → 0

Here the maps involving π₀ are not group homomorphisms because the π₀ are not groups, but they are exact in the sense that the image equals the kernel.

Example: the Hopf fibration. Let B equal S² and E equal S³. Let p be the Hopf fibration, which has fiber S¹. From the long exact sequence

… → π_n(S¹) → π_n(S³) → π_n(S²) → π_n−1(S¹) → …

and the fact that π_n(S¹) = 0 for n ≥ 2, we find that π_n(S³) = π_n(S²) for n ≥ 3. In particular, π₃(S²) = π₃(S³) = Z.

Relative homotopy groups

There are also relative homotopy groups π_n(X,A) for a pair (X,A). The elements of such a group are relative homotopy classes of maps Sⁿ → X. Two maps f, g are called homotopic relative to A if they are homotopic by a homotopy F : Sⁿ × [0,1] → X such that, for each a in A, the map F(a,t) is constant (independent of t). The ordinary homotopy groups are the special case in which A is the base point.

There is a long exact sequence of relative homotopy groups.

See also

• Knot theory