Homotopy group

Last updated

In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.

Contents

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.

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

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.

A torus Torus.png
A torus
A sphere 2sphere 2.png
A sphere

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't have the same topological structurea 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 T 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.

Definition

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, we can define πn(X) 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 Homotopy group addition.svg
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.

Long exact sequence of a fibration

Let p: EB be a basepoint-preserving Serre fibration with fiber F, 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 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

and the fact that πn(S1) = 0 for n ≥ 2, we find that πn(S3) = πn(S2) for n ≥ 3. In particular,

In the case of a cover space, when the fiber is discrete, we have that πn(E) is isomorphic to πn(B) for n > 1, that πn(E) embeds injectively into πn(B) for all positive n, and that the subgroup of π1(B) that corresponds to the embedding of π1(E) 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 [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 bundles

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 space

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 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. [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 S2 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 groups

Relative homotopy groups

There is also a useful generalization of homotopy groups, , called relative homotopy groups for a pair , where A is a subspace of X.

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 f, g are called homotopic relative toA if they are homotopic by a basepoint-preserving homotopy F : Dn × [0,1] → X such that, for each p in Sn−1 and t in [0,1], the element F(p,t) is in A. Note that ordinary homotopy groups are recovered for the special case in which is the base point.

These groups are abelian for n ≥ 3 but for n = 2 form the top group of a crossed module with bottom group π1(A).

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.

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 X, its n-th homotopy group is usually denoted by , and its n-th homology group is usually denoted by .

See also

Notes

  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".

Related Research Articles

In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent have isomorphic fundamental groups.

Homotopy Continuous deformation between two continuous maps

In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology.

Covering space

In mathematics, specifically algebraic topology, a covering map is a continuous function from a topological space to a topological space such that each point in has an open neighbourhood evenly covered by . In this case, is called a covering space and the base space of the covering projection. The definition implies that every covering map is a local homeomorphism.

In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X × G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equipped with

  1. An action of G on P, analogous to (x, g)h = for a product space.
  2. A projection onto X. For a product space, this is just the projection onto the first factor, (x,g) ↦ x.

In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space. A fibration is like a fiber bundle, except that the fibers need not be the same space, nor even homeomorphic; rather, they are just homotopy equivalent. Weak fibrations discard even this equivalence for a more technical property.

In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by Raoul Bott, which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres. Bott periodicity can be formulated in numerous ways, with the periodicity in question always appearing as a period-2 phenomenon, with respect to dimension, for the theory associated to the unitary group. See for example topological K-theory.

In mathematics, specifically in homotopy theory, a classifying spaceBG of a topological group G is the quotient of a weakly contractible space EG by a proper free action of G. It has the property that any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle EGBG. As explained later, this means that classifying spaces represent a set-valued functor on the homotopy category of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on the category of topological spaces, such as Sierpiński space. This notion is generalized by the notion of classifying topos. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy.

In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. This means given a cohomology theory

In mathematics, and algebraic topology in particular, an Eilenberg–MacLane space is a topological space with a single nontrivial homotopy group. As such, an Eilenberg–MacLane space is a special kind of topological space that can be regarded as a building block for homotopy theory; general topological spaces can be constructed from these via the Postnikov system. These spaces are important in many contexts in algebraic topology, including constructions of spaces, computations of homotopy groups of spheres, and definition of cohomology operations. The name is for Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s.

Homotopy groups of spheres

In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute.

In mathematics, the classifying space for the unitary group U(n) is a space BU(n) together with a universal bundle EU(n) such that any hermitian bundle on a paracompact space X is the pull-back of EU(n) by a map X → BU(n) unique up to homotopy.

In mathematics, stable homotopy theory is that part of homotopy theory concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the Freudenthal suspension theorem, which states that given any pointed space , the homotopy groups stabilize for sufficiently large. In particular, the homotopy groups of spheres stabilize for . For example,

In mathematics, the Serre spectral sequence is an important tool in algebraic topology. It expresses, in the language of homological algebra, the singular (co)homology of the total space X of a (Serre) fibration in terms of the (co)homology of the base space B and the fiber F. The result is due to Jean-Pierre Serre in his doctoral dissertation.

In mathematics, the Adams spectral sequence is a spectral sequence introduced by J. Frank Adams (1958) which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now called stable homotopy theory. It is a reformulation using homological algebra, and an extension, of a technique called 'killing homotopy groups' applied by the French school of Henri Cartan and Jean-Pierre Serre.

In mathematics, the Puppe sequence is a construction of homotopy theory, so named after Dieter Puppe. It comes in two forms: a long exact sequence, built from the mapping fibre, and a long coexact sequence, built from the mapping cone. Intuitively, the Puppe sequence allows us to think of homology theory as a functor that takes spaces to long-exact sequences of groups. It is also useful as a tool to build long exact sequences of relative homotopy groups.

In homotopy theory, a branch of algebraic topology, a Postnikov system is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree agrees with the truncated homotopy type of the original space . Postnikov systems were introduced by, and are named after, Mikhail Postnikov.

In mathematics, a weak equivalence is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the axiomatic definition of a model category.

This is a glossary of properties and concepts in algebraic topology in mathematics.

In mathematics, homotopy theory is a systematic study of situations in which maps come with homotopies between them. It originated as a topic in algebraic topology but nowadays it is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry (e.g., A1 homotopy theory) and category theory (specifically the study of higher categories).

References