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.

The "n"-dimensional unit sphere — called the "n"-sphere for brevity, and denoted as "S"^"n" — generalizes the familiar circle ("S"¹) and the ordinary sphere ("S"²). The "n"-sphere may be defined geometrically as the set of points in a Euclidean space of dimension "n" + 1 located at a unit distance from the origin. The "i"-th "homotopy group" π_"i"("S"^"n") summarizes the different ways in which the "i"-dimensional sphere "S"^"i" can be mapped continuously into the "n"-dimensional sphere "S"^"n". This summary does not distinguish between two mappings if one can be continuously deformed to the other; thus, only equivalence classes of mappings are summarized. An "addition" operation defined on these equivalence classes makes the set of equivalence classes into an abelian group.

The problem of determining π_"i"("S"^"n") falls into three regimes, depending on whether "i" is less than, equal to, or greater than "n". For 0 < "i" < "n", any mapping from "S"^"i" to "S"^"n" is topologically equivalent (i.e., can be reduced) to a one-point mapping, which maps all of "S"^"i" to a single point of "S"^"n". When "i" = "n", every map from "S"^"n" to itself has a degree that measures how many times the sphere is wrapped around itself. This degree identifies π_"n"("S"^"n") with the group of integers under addition. For example, every point on a circle can be mapped continuously onto a point of another circle; as the first point is moved around the first circle, the second point may cycle several times around the second circle, depending on the particular mapping. However, the most interesting and surprising results occur when "i" > "n". The first such surprise was the discovery of a mapping called the Hopf fibration, which wraps the 3-sphere "S"³ around the usual sphere "S"² in a non-trivial fashion, and so is not equivalent to a one-point mapping.

The question of computing the homotopy group π_"n"+"k"("S"^"n") for positive "k" turned out to be a central question in algebraic topology that has contributed to development of many of its fundamental techniques and has served as a stimulating focus of research. One of the main discoveries is that the homotopy groups π_"n"+"k"("S"^"n") are independent of "n" for "n" ≥ "k" + 2. These are called the stable homotopy groups of spheres and have been computed for values of "k" up to 64. The stable homotopy groups form the coefficient ring of an extraordinary cohomology theory, called stable cohomotopy theory. The unstable homotopy groups (for "n" < "k" + 2) are more erratic; nevertheless, they have been tabulated for "k" &lt; 20. Most modern computations use spectral sequences, a technique first applied to homotopy groups of spheres by Jean-Pierre Serre. Several important patterns have been established, yet much remains unknown and unexplained.

Background

The study of homotopy groups of spheres builds on a great deal of background material, here briefly reviewed. Algebraic topology provides the larger context, itself built on topology and abstract algebra, with homotopy groups as a basic example.

"n"-sphere

An ordinary sphere in three-dimensional space — the surface, not the solid ball — is just one example of what a sphere means in topology. Geometry defines a sphere rigidly, as a shape. Here are some alternatives.
* Implicit surface: "x"₀² + "x"₁² + "x"₂² = 1: This is the set of points in 3-dimensional Euclidean space found exactly one unit away from the origin. It is called the 2-sphere, "S"², for reasons given below. The same idea applies for any dimension "n"; the equation "x"₀² + "x"₁² + ⋯ + "x"_"n"² = 1 produces the "n"-sphere as a geometric object in ("n" + 1)-dimensional space. For example, the 1-sphere "S"¹ is a circle.
* Disk with collapsed rim: written in topology as "D"²/"S"¹: This construction moves from geometry to pure topology. The disk "D"² is the region contained by a circle, described by the inequality "x"₀² + "x"₁² ≤ 1, and its rim (or "boundary") is the circle "S"¹, described by the equality "x"₀² + "x"₁² = 1. If a balloon is punctured and spread flat it produces a disk; this construction repairs the puncture, like pulling a drawstring. The slash, pronounced "modulo", means to take the topological space on the left (the disk) and in it join together as one all the points on the right (the circle). The region is 2-dimensional, which is why topology calls the resulting topological space a 2-sphere. Generalized, "D"^"n"/"S"^"n"−1 produces "S"^"n". For example, "D"¹ is a line segment, and the construction joins its ends to make a circle. An equivalent description is that the boundary of an "n"-dimensional disk is glued to a point, producing a CW complex.
* Suspension of equator: written in topology as Σ"S"¹: This construction, though simple, is of great theoretical importance. Take the circle "S"¹ to be the equator, and sweep each point on it to one point above (the North Pole), producing the northern hemisphere, and to one point below (the South Pole), producing the southern hemisphere. For each positive integer "n", the "n"-sphere "x"₀² + "x"₁² + ⋯ + "x"_"n"² = 1 has as equator the ("n" − 1)-sphere "x"₀² + "x"₁² + ⋯ + "x"_"n"−1² = 1, and the suspension Σ"S"^"n"−1 produces "S"^"n".

Some theory requires selecting a fixed point on the sphere, calling the pair (sphere, point) a "pointed sphere". For some spaces the choice matters, but for a sphere all points are equivalent so the choice is a matter of convenience. The point (1, 0, 0, …, 0), which is on the equator of all the spheres, works well for geometric spheres; the (collapsed) rim of the disk is another obvious choice.

Homotopy group

The distinguishing feature of a topological space is its continuity structure, formalized in terms of open sets or neighborhoods. A continuous map is a function between spaces that preserves continuity. A homotopy is a continuous path between continuous maps; two maps connected by a homotopy are said to be homotopic. The idea common to all these concepts is to discard variations that do not affect outcomes of interest. An important practical example is the residue theorem of complex analysis, where "closed curves" are continuous maps from the circle into the complex plane, and where two closed curves produce the same integral result if they are homotopic in the topological space consisting of the plane minus the points of singularity.

The first homotopy group, or fundamental group, π₁("X") of a (path connected) topological space "X" thus begins with continuous maps from a pointed circle ("S"¹,"s") to the pointed space ("X","x"), where maps from one pair to another map "s" into "x". These maps (or equivalently, closed curves) are grouped together into equivalence classes based on homotopy (keeping the "base point" "x" fixed), so that two maps are in the same class if they are homotopic. Just as one point is distinguished, so one class is distinguished: all maps (or curves) homotopic to the constant map "S"¹↦"x" are called null homotopic. The classes become an abstract algebraic group with the introduction of addition, defined via an "equator pinch". This pinch maps the equator of a pointed sphere (here a circle) to the distinguished point, producing a "bouquet of spheres" — two pointed spheres joined at their distinguished point. The two maps to be added map the upper and lower spheres separately, agreeing on the distinguished point, and composition with the pinch gives the sum map.

More generally, the "i"-th homotopy group, π_"i"("X") begins with the pointed "i"-sphere ("S"^"i","s"), and otherwise follows the same procedure. The null homotopic class acts as the identity of the group addition, and for "X" equal to "S"^"n" (for positive "n") — the homotopy groups of spheres — the groups are abelian and finitely generated. If for some "i" all maps are null homotopic, then the group π_"i" consists of one element, and is called the trivial group.

A continuous map between two topological spaces induces a group homomorphism between the associated homotopy groups. In particular, if the map is a continuous bijection (a homeomorphism), so that the two spaces have the same topology, then their "i"-th homotopy groups are isomorphic for all "i". However, the real plane has exactly the same homotopy groups as a solitary point (as does a Euclidean space of any dimension), and the real plane with a point removed has the same groups as a circle, so groups alone are not enough to distinguish spaces. Although the loss of discrimination power is unfortunate, it can also make certain computations easier.

Low-dimensional examples

The low-dimensional examples of homotopy groups of spheres provide a sense of the subject, because these special cases can be visualized in ordinary 3-dimensional space harv|Hatcher|2002. However, such visualizations are not mathematical proofs, as they do not capture the possible complexity of maps between spheres.

= &pi;₁("S"¹) = Z=

The simplest case concerns the ways that a circle (1-sphere) can be wrapped around another circle. This can be visualized by wrapping a rubber band around one's finger: it can be wrapped once, twice, three times and so on. The wrapping can be in either of two directions, and wrappings in opposite directions will cancel out after a deformation. The homotopy group π₁("S"¹) is therefore an infinite cyclic group, and is isomorphic to the group Z of integers under addition: a homotopy class is identified with an integer by counting the number of times a mapping in the homotopy class wraps around the circle. This integer can also be thought of as the winding number of a loop around the origin in the plane.

The identification (a group isomorphism) of the homotopy group with the integers is often written as an equality: thus π₁("S"¹) = Z.

=&pi;₂("S"²) = Z=

Mappings from a 2-sphere to a 2-sphere can be visualized as wrapping a plastic bag around a ball and then sealing it. The sealed bag is topologically equivalent to a 2-sphere, as is the surface of the ball. The bag can be wrapped more than once by twisting it and wrapping it back over the ball. (There is no requirement for the continuous map to be injective and so the bag is allowed to pass through itself.) The twist can be in one of two directions and opposite twists can cancel out by deformation. The total number of twists after cancellation is an integer, called the "degree" of the mapping. As in the case mappings from the circle to the circle, this degree identifies the homotopy group with the group of integers, Z.

These two results generalize: for all "n" > 0, π_"n"("S"^"n") = Z (see below).

= &pi;₁("S"²) = 0=

Any continuous mapping from a circle to an ordinary sphere can be continuously deformed to a one-point mapping, and so its homotopy class is trivial. One way to visualize this is to imagine a rubber-band wrapped around a frictionless ball: the band can always be slid off the ball. The homotopy group is therefore a trivial group, with only one element, the identity element, and so it can be identified with the subgroup of Z consisting only of the number zero. This group is often denoted by 0.

This result generalises to higher dimensions. All mappings from a lower-dimensional sphere into a sphere of higher dimension are similarly trivial: if "i" &lt; "n", then π_"i"("S"^"n") = 0.

= &pi;₂("S"¹) = 0=

All the interesting cases of homotopy groups of spheres involve mappings from a higher-dimensional sphere onto one of lower dimension. Unfortunately, the only example which can easily be visualized is not interesting: there are no nontrivial mappings from the ordinary sphere to the circle. Hence, π₂("S"¹) = 0.

= &pi;₃("S"²) = Z=

The first nontrivial example with "i" > "n" concerns mappings from the 3-sphere to the ordinary 2-sphere, and was discovered by Heinz Hopf, who constructed a nontrivial map from "S"³ to "S"², now known as the Hopf fibration harv|Hopf|1931. This map generates the homotopy group π₃("S"²) and π₃("S"²) = Z.

History

In the late 19th century Camille Jordan introduced the notion of homotopy and used the notion of a homotopy group, without using the language of group theory harv|O'Connor|Robertson|2001. A more rigorous approach was adopted by Henri Poincaré in his 1895 set of papers "Analysis situs" where the related concepts of homology and the fundamental group were also introduced harv|O'Connor|Robertson|1996.

Higher homotopy groups were first defined by Eduard Čech in 1932 harv|Čech|1932|p=203. (His first paper was withdrawn on the advice of Pavel Sergeyevich Alexandrov and Heinz Hopf, on the grounds that the groups were commutative so could not be the right generalizations of the fundamental group.) Witold Hurewicz is also credited with the introduction of homotopy groups in his 1935 paper and also for the Hurewicz theorem which can be used to calculate some of the groups harv|May|1999a.An important method for calculating the various groups is the concept of stable algebraic topology, which finds properties that are independent of the dimensions. Typically these only hold for larger dimensions. The first such result was Hans Freudenthal's suspension theorem, published in 1937. Stable algebraic topology flourished between 1945 and 1966 with many important results harv|May|1999a. In 1953 George W. Whitehead showed that there is a metastable range for the homotopy groups of spheres. Jean-Pierre Serre used spectral sequences to show that most of these groups are finite, the exceptions being π_"n"("S"^"n") and π_4"n"−1("S"^2"n"). Others who worked in this area included José Ádem, J. Peter May, Hiroshi Toda and Lev Semenovich Pontryagin. The stable homotopy groups π_"n"+"k"("S"^"n") are known for "k" up to 64, and, as of 2007, unknown for larger "k" harv|Hatcher|2002|loc=Stable homotopy groups, pp. 385–393.

General theory

As noted already, when "i" is less than "n", π_"i"("S"^"n") = 0, the trivial group harv|Hatcher|2002. The reason is that a continuous mapping from an "i"-sphere to an "n"-sphere with "i" < "n" can always be deformed so that it is not surjective. Consequently, its image is contained in "S"^"n" with a point removed; this is a contractible space, and any mapping to such a space can be deformed into a one-point mapping.

The case "i" = "n" has also been noted already, and is an easy consequence of the Hurewicz theorem: this theorem links homotopy groups with homology groups, which are generally easier to calculate; in particular, it shows that for a simply-connected space "X", the first nonzero homotopy group π_"k"("X"), with "k" &gt; 0, is isomorphic to the first nonzero homology group "H"_"k"("X"). For the "n"-sphere, this immediately implies that for "n" &gt; 0, π_"n"("S"^"n") = "H"_"n"("S"^"n") = Z.

The homology groups "H"_"i"("S"^"n"), with "i" &gt; "n", are all trivial. It therefore came as a great surprise historically that the corresponding homotopy groups are not trivial in general. This is the case that is of real importance: the higher homotopy groups π_"i"("S"^"n"), for "i" &gt; "n", are surprisingly complex and difficult to compute, and the effort to compute them has generated a significant amount of new mathematics.

The following table of gives an idea of the complexity of the higher homotopy groups even for spheres of dimension 8 or less. In this table, the entries are either the trivial group 0, the infinite cyclic group Z, finite cyclic groups of order "n" (written as Z_n), or direct products of such groups (written, for example, as Z₂₄&times;Z₃ or Z₂² = Z₂&times;Z₂). Extended tables of homotopy groups of spheres are given at the end of the article.

Table of stable homotopy groups

The stable homotopy groups π_"k" are the product of cyclic groups of the infinite or prime power ordersshown in the table. (For largely historical reasons, stable homotopy groups are usually given as products of cyclic groups of prime power order, while tables of unstable homotopy groups often give them as products of the smallest number of cyclic groups.) The main complexity is in the 2-, 3-, and 5-components: for "p" > 5, the "p"-components in the range of the table are accounted for by the J-homomorphism and are cyclic of order "p" if 2("p"−1) divides "k"+1 and 0 otherwise harv|Fuks|2001. (The 2-components can be found in harvtxt|Kochman|1990, and the 3- and 5-components in harvtxt|Ravenel|2003.) The mod 8 behavior of the table comes from Bott periodicity via the J-homomorphism, whose image is underlined. For "k" = 54, 62, or 63 there is uncertainty in the structure of the 2-component.

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