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.
Image:Hopf Fibration.png|right|thumb|The Hopf fibration is a nontrivial mapping of the 3-sphere to the 2-sphere, and generates the third homotopy group of the 2-sphere.
Image:Hopfkeyrings.jpg|right|thumb|This picture mimics part of the Hopf fibration, an interesting mapping from the three-dimensional sphere to the two-dimensional sphere. This mapping is the generator of the third homotopy group of the 2-sphere.
The -dimensional unit sphere — called the -sphere for brevity, and denoted as — generalizes the familiar circle and the ordinary sphere. The -sphere may be defined geometrically as the set of points in a Euclidean space of dimension located at a unit distance from the origin. The -th homotopy group summarizes the different ways in which the -dimensional sphere can be mapped continuously into the sphere. 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 falls into three regimes, depending on whether is less than, equal to, or greater than :

For, any mapping from to is homotopic to a constant mapping, i.e., a mapping that maps all of to a single point of. In the smooth case, it follows directly from Sard's Theorem. Therefore the homotopy group is the trivial group.
When, every map from to itself has a degree that measures how many times the sphere is wrapped around itself. This degree identifies the homotopy group 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.
The most interesting and surprising results occur when. The first such surprise was the discovery of a mapping called the Hopf fibration, which wraps the 3-sphere around the usual sphere in a non-trivial fashion, and so is not equivalent to a one-point mapping.

The question of computing the homotopy group for positive 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 are independent of for. These are called the stable homotopy groups of spheres and have been computed for values of up to 90. The stable homotopy groups form the coefficient ring of an extraordinary cohomology theory, called stable cohomotopy theory. The unstable homotopy groups are more erratic; nevertheless, they have been tabulated for . 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.

-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:
Disk with collapsed rim: written in topology as
Suspension of equator: written in topology as

Some theory requires selecting a fixed point on the sphere, calling the pair 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. For spheres constructed as a repeated suspension, the point, which is on the equator of all the levels of suspension, works well; for the disk with collapsed rim, the point resulting from the collapse of the rim 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, of a topological space thus begins with continuous maps from a pointed circle to the pointed space, where maps from one pair to another map into. These maps are grouped together into equivalence classes based on homotopy, 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 homotopic to the constant map 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 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 -th homotopy group, begins with the pointed -sphere, and otherwise follows the same procedure. The null homotopic class acts as the identity of the group addition, and for equal to — the homotopy groups of spheres — the groups are abelian and finitely generated. If for some all maps are null homotopic, then the group 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, so that the two spaces have the same topology, then their -th homotopy groups are isomorphic for all. However, the real plane has exactly the same homotopy groups as a solitary point, 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. However, such visualizations are not mathematical proofs, and do not capture the possible complexity of maps between spheres.

The simplest case concerns the ways that a circle 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 is therefore an infinite cyclic group, and is isomorphic to the group 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 of the homotopy group with the integers is often written as an equality: thus.

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. 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,.
These two results generalize: for all, .

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 consisting only of the number zero. This group is often denoted by 0. Showing this rigorously requires more care, however, due to the existence of
space-filling curves.
This result generalizes to higher dimensions. All mappings from a lower-dimensional sphere into a sphere of higher dimension are similarly trivial: if, then . This can be shown as a consequence of the cellular approximation theorem.