Homotopy group

Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.

To define the n-th homotopy group, the base-point-preserving maps from an n-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes.

Topological spaces with differing homotopy groups are never homeomorphic, but topological spaces that are not homeomorphic can have the same homotopy groups.

The notion of homotopy of paths was introduced by Camille Jordan.

However, since continuity (the basic notion of topology) only deals with the local structure, it can be difficult to formally define the obvious global difference.

The homotopy groups, however, carry information about the global structure.

because the universal cover of the torus is the Euclidean plane

Here the quotient is in the category of topological spaces, rather than groups or rings.

In particular, the equivalence classes are given by homotopies that are constant on the basepoint of the sphere.

The idea of composition in the fundamental group is that of traveling 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.

For the corresponding definition in terms of spheres, define the sum

[2] Further, similar to the fundamental group, for a path-connected space any two choices of basepoint give rise to isomorphic

[3] It is tempting to try to simplify the definition of homotopy groups by omitting the base points, but this does not usually work for spaces that are not simply connected, even for path-connected spaces.

This holds if-and-only-if the d-th homotopy group of X is not trivial.

In short, X has a hole with a d-dimensional boundary, if-and-only-if

that is, a map possessing the homotopy lifting property with respect to CW complexes.

Then there is a long exact sequence of homotopy groups

are not groups, but they are exact in the sense that the image equals the kernel.

When the fibration is the mapping fibre, or dually, the cofibration is the mapping cone, then the resulting exact (or dually, coexact) sequence is given by the Puppe sequence.

There are many realizations of spheres as homogeneous spaces, which provide good tools for computing homotopy groups of Lie groups, and the classification of principal bundles on spaces made out of spheres.

which can be computed using the Postnikov system, we have the long exact sequence

Unlike the Seifert–van Kampen theorem for the fundamental group and the excision theorem for singular homology and cohomology, there is no simple known way to calculate the homotopy groups of a space by breaking it up into smaller spaces.

See for a sample result the 2010 paper by Ellis and Mikhailov.

However, despite intense research in calculating the homotopy groups of spheres, even in two dimensions a complete list is not known.

In particular the Serre spectral sequence was constructed for just this purpose.

Hence, we have the following construction: The elements of such a group are homotopy classes of based maps

Note that ordinary homotopy groups are recovered for the special case in which

There is also a long exact sequence of relative homotopy groups that can be obtained via the Puppe sequence: The homotopy groups are fundamental to homotopy theory, which in turn stimulated the development of model categories.

It is possible to define abstract homotopy groups for simplicial sets.