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In the sphere Sn we choose a base point a. For a space X with base point b, we define πn(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 πn(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, … tn) = f(2t1, t2, … tn) for t1 in [0,1/2] and (f + g)(t1, t2, … tn) = g(2t1-1, t2, … tn) 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 πn is abelian. (For a proof of this, note that in two dimensions or greater, two homotopies can be "rotated" around each other.)
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 complexIn topology, a CW complex is a type of topological space introduced by J. Whitehead to meet the needs of homotopy theory. The idea was to have a class of spaces that was broader than simplicial complexes (we could say now, had better categorical propertiees. Then there is a long exact sequenceHomological algebra In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the i of homotopy groups
Here the maps involving π0 are not group homomorphismThis word should not be confused with homeomorphism. In abstract algebra, a homomorphism is a map from one algebraic structure to another of the same type that preserves all the relevant structure. Some authors use the word homomorphism in a larger contexs 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
and the fact that πn(S1) = 0 for n ≥ 2, we find that πn(S3) = πn(S2) for n ≥ 3. In particular, π3(S2) = π3(S3) = Z.