Homotopy group

Summary

In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted 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, 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.

The notion of homotopy of paths was introduced by Camille Jordan.[1]

IntroductionEdit

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.

DefinitionEdit

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.

If   then   is abelian.[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. 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  .

Long exact sequence of a fibrationEdit

Let   be a basepoint-preserving Serre fibration with fiber   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

 

Here the maps involving   are not group homomorphisms because the   are not groups, but they are exact in the sense that the image equals the kernel.

Example: the Hopf fibration. Let B equal   and E equal   Let p be the Hopf fibration, which has fiber   From the long exact sequence

 

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.

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 spheresEdit

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 groupEdit

There is a fibration[4]

 

giving the long exact sequence

 

which computes the low order homotopy groups of   for   since   is  -connected. In particular, there is a fibration

 

whose lower homotopy groups can be computed explicitly. Since   and there is the fibration

 

we have   for   Using this, and the fact that   which can be computed using the Postnikov system, we have the long exact sequence

 

Since   we have   Also, the middle row gives   since the connecting map   is trivial. Also, we can know   has two-torsion.

Application to sphere bundlesEdit

Milnor[5] used the fact   to classify 3-sphere bundles over   in particular, he was able to find exotic spheres which are smooth manifolds called Milnor's spheres only homeomorphic to   not diffeomorphic. Note that any sphere bundle can be constructed from a  -vector bundle, which have structure group   since   can have the structure of an oriented Riemannian manifold.

Complex projective spaceEdit

There is a fibration

 

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.[6]

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.

Certain homotopy groups of n-connected spaces can be calculated by comparison with homology groups via the Hurewicz theorem.

A list of methods for calculating homotopy groupsEdit

Relative homotopy groupsEdit

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 to A 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:

 

Related notionsEdit

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 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".[7] Given a topological space   its n-th homotopy group is usually denoted by   and its n-th homology group is usually denoted by  

See alsoEdit

NotesEdit

  1. ^ Marie Ennemond Camille Jordan
  2. ^ For a proof of this, note that in two dimensions or greater, two homotopies can be "rotated" around each other. See Eckmann–Hilton argument.
  3. ^ see Allen Hatcher#Books section 4.1.
  4. ^ Husemoller. Fiber Bundles. p. 89.
  5. ^ Milnor, John (1956). "On manifolds homeomorphic to the 7-sphere". Annals of Mathematics. 64: 399–405.
  6. ^ Ellis, Graham J.; Mikhailov, Roman (2010). "A colimit of classifying spaces". Advances in Mathematics. 223 (6): 2097–2113. arXiv:0804.3581. doi:10.1016/j.aim.2009.11.003. MR 2601009.
  7. ^ Wildberger, N. J. (2012). "An introduction to homology". Archived from the original on 2021-12-12.

ReferencesEdit

  • Ronald Brown, `Groupoids and crossed objects in algebraic topology', Homology, Homotopy and Applications, 1 (1999) 1–78.
  • Ronald Brown, Philip J. Higgins, Rafael Sivera, Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics Vol. 15, 703 pages, European Math. Society, Zürich, 2011. doi:10.4171/083 MR2841564
  • Čech, Eduard (1932), "Höherdimensionale Homotopiegruppen", Verhandlungen des Internationalen Mathematikerkongress, Zürich.
  • Hatcher, Allen (2002), Algebraic topology, Cambridge University Press, ISBN 978-0-521-79540-1
  • "Homotopy group", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
  • Hopf, Heinz (1931), "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche", Mathematische Annalen, 104 (1): 637–665, doi:10.1007/BF01457962.
  • Kamps, Klaus H.; Porter, Timothy (1997). Abstract homotopy and simple homotopy theory. River Edge, NJ: World Scientific Publishing. doi:10.1142/9789812831989. ISBN 981-02-1602-5. MR 1464944.
  • Toda, Hiroshi (1962). Composition methods in homotopy groups of spheres. Annals of Mathematics Studies. Vol. 49. Princeton, N.J.: Princeton University Press. ISBN 0-691-09586-8. MR 0143217.
  • Whitehead, George William (1978). Elements of homotopy theory. Graduate Texts in Mathematics. Vol. 61 (3rd ed.). New York-Berlin: Springer-Verlag. pp. xxi+744. ISBN 978-0-387-90336-1. MR 0516508.