Homotopy group
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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.
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[edit] Homotopy groups
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 classes 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.)
[edit] 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
- … → πn(F) → πn(E) → πn(B) → πn−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
- … → πn(S1) → πn(S3) → πn(S2) → πn−1(S1) → …
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.
[edit] Methods of Calculation
Calculation of homotopy groups is in general much more difficult than some of the other homotopy invariants learned in algebraic topology. Unlike the Seifert-van Kampen theorem for the fundamental group and the Excision theorem for singular homology and cohomology, there is no simple way to calculate the homotopy groups of a space by breaking it up into smaller spaces.
For some spaces, such as tori, all higher homotopy groups (that is, second and higher homotopy groups) are trivial. These are the so-called aspherical spaces, and it is an interesting fact that as one increases in dimension the frequency of aspherical manifolds increases. However, despite intense research in calculating the homotopy groups of spheres, even in two dimensions a complete list is not known. To calculate even the fourth homotopy group of S2 one needs much more advanced techniques than the definitions might suggest. In particular the Serre spectral sequence was constructed for just this purpose.
[edit] Relative homotopy groups
There are also relative homotopy groups πn(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.
[edit] As a functor
The relation between the category of topological spaces and the category of groups or the category of topological pair into the category of long exact sequences of spaces.
[edit] History
The notion of homotopy of paths was introduced by Camille Jordan. [1]
