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.
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 captures the different ways in which the -dimensional sphere can be mapped continuously into the sphere . It does not distinguish between mappings that can be continuously deformed into one another; its elements are therefore equivalence classes of maps under homotopy. An addition operation makes this set of equivalence classes into an abelian group whose identity element is the class of any constant map, i.e. a one that maps all of to a single point of .
The problem of determining falls into three regimes, depending on whether is less than, equal to, or greater than : For , any map from to is homotopic (i.e., continuously deformable) to a constant map, so is the trivial group. This is a consequence of the cellular approximation theorem. When , every map from to itself can be assigned a degree that intuitively measures how many times the sphere is wrapped around itself. This degree identifies the homotopy group with the group of integers under addition. For , the groups depend strongly on and . The historically first interesting example is the group which is generated by the Hopf fibration.
The problem of computing the groups for positive is very difficult and of central importance to the field of algebraic topology, and as such has motivated the development of many fundamental tools and techniques. One early important result is the Freudenthal suspension theorem, which states that is independent of for . These groups, known as the stable homotopy groups of spheres, can be studied using the tools of stable homotopy theory and are completely known for values of up to 90.. The groups in the unstable range are less accessible, but have been tabulated at least for all .
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: 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, , for reasons given below. The same idea applies for any dimension ; the equation produces the -sphere as a geometric object in ()-dimensional space. For example, the 1-sphere is a circle. Disk with collapsed rim: written in topology as This construction moves from geometry to pure topology. The disk is the region contained by a circle, described by the inequality , and its rim (or "boundary") is the circle , described by the equality . 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, produces . For example, is a line segment, and the construction joins its ends to make a circle. An equivalent description is that the boundary of an -dimensional disk is glued to a point, producing a CW complex. Suspension of equator: written in topology as This construction, though simple, is of great theoretical importance. Take the circle 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 , the -sphere has as equator the ()-sphere , and the suspension produces .
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 (path connected) 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 (or equivalently, closed curves) are grouped together into equivalence classes based on homotopy (keeping the "base point" 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 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 -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 (for positive ) — 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 that has a continuous inverse function (a homeomorphism), 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 (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. 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 (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 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 (a group isomorphism) of the homotopy group with the integers is often written as an equality: thus .See, e.g., , Section 8.1, "".
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, .
These two results generalize: for all , (see below).
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.
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, . This is because has the real line as its universal cover which is contractible (it has the homotopy type of a point). In addition, because is simply connected, by the lifting criterion, any map from to can be lifted to a map into the real line and the nullhomotopy descends to the downstairs space (via composition).
The first nontrivial example with concerns mappings from the 3-sphere to the ordinary 2-sphere, and was discovered by Heinz Hopf, who constructed a nontrivial map from to , now known as the Hopf fibration. This map generates the homotopy group .
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. 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.