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. Two mappings are homotopic if one can be continuously deformed into the other. These homotopy classes form a group, called then-th homotopy group, of the given space X with base point. Topological spaces with differing homotopy groups are never equivalent (homeomorphic), but topological spaces that are not homeomorphic can have the same homotopy groups.
In modern mathematics it is common to study a category by associating to every object of this category a simpler object that still retains sufficient information about the object of interest. Homotopy groups are such a way of associating groups to topological spaces.
That link between topology and groups lets mathematicians apply insights from group theory to topology. For example, if two topological objects have different homotopy groups, they can not have the same topological structure—a fact that may be difficult to prove using only topological means. For example, the torus is different from the sphere: the torus has a "hole"; the sphere doesn't. 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.
As for the example: the first homotopy group of the torus is
because the universal cover of the torus is the Euclidean plane mapping to the torus Here the quotient is in the category of topological spaces, rather than groups or rings. On the other hand, the sphere satisfies:
because every loop can be contracted to a constant map (see homotopy groups of spheres for this and more complicated examples of homotopy groups). Hence the torus is not homeomorphic to the sphere.
In the n-sphere we choose a base point a. For a space X with base point b, we define to be the set of homotopy classes of maps
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, define to be the group of homotopy classes of maps from the n-cube to X that take the boundary of the n-cube to b.
Composition in the fundamental group
For the homotopy classes form a group. To define the group operation, recall that in the fundamental group, the product of two loops is defined by setting
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. We therefore define the sum of maps by the formula
For the corresponding definition in terms of spheres, define the sum of maps to be composed with h, where is the map from 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.
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. The set of homotopy classes of maps from a sphere to a path connected space is not the homotopy group, but is essentially the set of orbits of the fundamental group on the homotopy group, and in general has no natural group structure.
A way out of these difficulties has been found by defining higher homotopy groupoids of filtered spaces and of n-cubes of spaces. These are related to relative homotopy groups and to n-adic homotopy groups respectively. A higher homotopy van Kampen theorem then enables one to derive some new information on homotopy groups and even on homotopy types. For more background and references, see "Higher dimensional group theory" and the references below.
Homotopy groups and holesEdit
A topological space has a hole with a d-dimensional boundary if-and-only-if it contains a d-dimensional sphere that cannot be shrunk continuously to a single point. This holds if-and-only-if there is a mapping that is not homotopic to a constant function. 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 .
and the fact that for we find that for In particular,
In the case of a cover space, when the fiber is discrete, we have that is isomorphic to for that embeds injectively into for all positive and that the subgroup of that corresponds to the embedding of has cosets in bijection with the elements of the fiber.
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.
where is the unit sphere in This sequence can be used to show the simple-connectedness of for all
Methods of calculationEdit
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 known way to calculate the homotopy groups of a space by breaking it up into smaller spaces. However, methods developed in the 1980s involving a van Kampen type theorem for higher homotopy groupoids have allowed new calculations on homotopy types and so on homotopy groups. See for a sample result the 2010 paper by Ellis and Mikhailov.
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. 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 one needs much more advanced techniques than the definitions might suggest. In particular the Serre spectral sequence was constructed for just this purpose.
There is also a useful generalization of homotopy groups, called relative homotopy groups for a pair where A is a subspace of
The construction is motivated by the observation that for an inclusion there is an induced map on each homotopy group which is not in general an injection. Indeed, elements of the kernel are known by considering a representative and taking a based homotopy to the constant map or in other words while the restriction to any other boundary component of is trivial. Hence, we have the following construction:
The elements of such a group are homotopy classes of based maps which carry the boundary into A. Two maps are called homotopic relative toA if they are homotopic by a basepoint-preserving homotopy such that, for each p in and t in the element is in A. Note that ordinary homotopy groups are recovered for the special case in which is the singleton containing the base point.
These groups are abelian for but for form the top group of a crossed module with bottom group
There is also a long exact sequence of relative homotopy groups that can be obtained via the Puppe sequence:
Homology groups are similar to homotopy groups in that they can represent "holes" in a topological space. However, homotopy groups are usually not commutative, and often very complex and hard to compute. In contrast, homology groups are commutative (as are the higher homotopy groups). Hence, it is sometimes said that "homology is a commutative alternative to homotopy". Given a topological space its n-th homotopy group is usually denoted by and its n-th homology group is usually denoted by