Algebraic construct classifying topological spaces
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted formula_1 which records information about loops in a space. 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. 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, formula_2 of the given space "X" with base point. 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.
Introduction.
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 cannot 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 formula_3 is
formula_4
because the universal cover of the torus is the Euclidean plane formula_5 mapping to the torus formula_6 Here the quotient is in the category of topological spaces, rather than groups or rings. On the other hand, the sphere formula_7 satisfies:
formula_8
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.
Definition.
In the "n"-sphere formula_9 we choose a base point "a". For a space "X" with base point "b", we define formula_10 to be the set of homotopy classes of maps
formula_11
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 formula_10 to be the group of homotopy classes of maps formula_13 from the "n"-cube to "X" that take the boundary of the "n"-cube to "b".
For formula_14 the homotopy classes form a group. To define the group operation, recall that in the fundamental group, the product formula_15 of two loops formula_16 is defined by setting
formula_17
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 formula_18 by the formula
formula_19
For the corresponding definition in terms of spheres, define the sum formula_20 of maps formula_21 to be formula_22 composed with "h", where formula_22 is the map from formula_9 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 formula_25 then formula_26 is abelian. Further, similar to the fundamental group, for a path-connected space any two choices of basepoint give rise to isomorphic formula_27
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 holes.
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 formula_28 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 formula_29.
Long exact sequence of a fibration.
Let formula_30 be a basepoint-preserving Serre fibration with fiber formula_31 that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that "B" is path-connected. Then there is a long exact sequence of homotopy groups
formula_32
Here the maps involving formula_33 are not group homomorphisms because the formula_33 are not groups, but they are exact in the sense that the image equals the kernel.
Example: the Hopf fibration. Let "B" equal formula_7 and "E" equal formula_36 Let "p" be the Hopf fibration, which has fiber formula_37 From the long exact sequence
formula_38
and the fact that formula_39 for formula_25 we find that formula_41 for formula_42 In particular, formula_43
In the case of a cover space, when the fiber is discrete, we have that formula_44 is isomorphic to formula_45 for formula_46 that formula_44 embeds injectively into formula_45 for all positive formula_49 and that the subgroup of formula_50 that corresponds to the embedding of formula_51 has cosets in bijection with the elements of the fiber.
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.
Homogeneous spaces and spheres.
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.
Special orthogonal group.
There is a fibration
formula_52
giving the long exact sequence
formula_53
which computes the low order homotopy groups of formula_54 for formula_55 since formula_56 is formula_57-connected. In particular, there is a fibration
formula_58
whose lower homotopy groups can be computed explicitly. Since formula_59 and there is the fibration
formula_60
we have formula_61 for formula_62 Using this, and the fact that formula_63 which can be computed using the Postnikov system, we have the long exact sequence
formula_64
Since formula_65 we have formula_66 Also, the middle row gives formula_67 since the connecting map formula_68 is trivial. Also, we can know formula_69 has two-torsion.
Application to sphere bundles.
Milnor used the fact formula_70 to classify 3-sphere bundles over formula_71 in particular, he was able to find exotic spheres which are smooth manifolds called Milnor's spheres only homeomorphic to formula_72 not diffeomorphic. Note that any sphere bundle can be constructed from a formula_73-vector bundle, which have structure group formula_74 since formula_75 can have the structure of an oriented Riemannian manifold.
Complex projective space.
There is a fibration
formula_76
where formula_77 is the unit sphere in formula_78 This sequence can be used to show the simple-connectedness of formula_79 for all formula_80
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 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 formula_7 one needs much more advanced techniques than the definitions might suggest. In particular the Serre spectral sequence was constructed for just this purpose.
Certain homotopy groups of "n"-connected spaces can be calculated by comparison with homology groups via the Hurewicz theorem.
Relative homotopy groups.
There is also a useful generalization of homotopy groups, formula_2 called relative homotopy groups formula_83 for a pair formula_84 where "A" is a subspace of formula_85
The construction is motivated by the observation that for an inclusion formula_86 there is an induced map on each homotopy group formula_87 which is not in general an injection. Indeed, elements of the kernel are known by considering a representative formula_88 and taking a based homotopy formula_89 to the constant map formula_90 or in other words formula_91 while the restriction to any other boundary component of formula_92 is trivial. Hence, we have the following construction:
The elements of such a group are homotopy classes of based maps formula_93 which carry the boundary formula_56 into "A". Two maps formula_95 are called homotopic relative to "A" if they are homotopic by a basepoint-preserving homotopy formula_96 such that, for each "p" in formula_56 and "t" in formula_98 the element formula_99 is in "A". Note that ordinary homotopy groups are recovered for the special case in which formula_100 is the singleton containing the base point.
These groups are abelian for formula_101 but for formula_102 form the top group of a crossed module with bottom group formula_103
There is also a long exact sequence of relative homotopy groups that can be obtained via the Puppe sequence:
formula_104
Related notions.
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.
Homology groups are similar to homotopy groups in that they can represent "holes" in a topological space. However, homotopy groups are 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 formula_105 its "n"-th homotopy group is usually denoted by formula_2 and its "n"-th homology group is usually denoted by formula_107
Notes.
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