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 -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 -dimensional 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 . 
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 64. The stable homotopy groups form the coefficient ring of an extraordinary cohomology theory, called stable cohomotopy theory. The unstable homotopy groups (for ) 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.
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. The point , 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, 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 (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 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 .
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 , (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 ℤ 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.
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 .
Higher homotopy groups were first defined by Eduard Čech in 1932 . (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 .
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 . 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 and . Others who worked in this area included José Adem, Hiroshi Toda, Frank Adams and J. Peter May. The stable homotopy groups are known for up to 64, and, as of 2007, unknown for larger .
General theory.
As noted already, when is less than , , the trivial group . The reason is that a continuous mapping from an -sphere to an -sphere with can always be deformed so that it is not surjective. Consequently, its image is contained in 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 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 , with , is isomorphic to the first nonzero homology group . For the -sphere, this immediately implies that for , .
The homology groups , with , 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 , for , are surprisingly complex and difficult to compute, and the effort to compute them has generated a significant amount of new mathematics.
Table.
The following table 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 (written as ), or direct products of such groups (written, for example, as or formula_1). Extended tables of homotopy groups of spheres are given at the end of the article.
The first two rows of this table are straightforward. The homotopy groups of the 0-dimensional sphere are trivial for , because any base point preserving map from an -sphere to a 0-sphere "is" a one-point mapping. Similarly, the homotopy groups of the 1-sphere are trivial for , because the universal covering space, ℝ, which has the same higher homotopy groups, is contractible.
Beyond these two rows, the higher homotopy groups () appear to be chaotic, but in fact there are many patterns, some obvious and some very subtle.
These patterns follow from many different theoretical results.
Stable and unstable groups.
The fact that the groups below the jagged line in the table above are constant along the diagonals is explained by the suspension theorem of Hans Freudenthal, which implies that the suspension homomorphism from to is an isomorphism for . The groups with are called the "stable homotopy groups of spheres", and are denoted : they are finite abelian groups for , and have been computed in numerous cases, although the general pattern is still elusive. . For , the groups are called the "unstable homotopy groups of spheres".
Hopf fibrations.
The classical Hopf fibration is a fiber bundle:
The general theory of fiber bundles shows that there is a long exact sequence of homotopy groups
For this specific bundle, each group homomorphism , induced by the inclusion , maps all of to zero, since the lower-dimensional sphere can be deformed to a point inside the higher-dimensional one . This corresponds to the vanishing of . Thus the long exact sequence breaks into short exact sequences,
Since is a suspension of , these sequences are split by the suspension homomorphism , giving isomorphisms
Since vanishes for at least 3, the first row shows that and are isomorphic whenever is at least 3, as observed above.
The Hopf fibration may be constructed as follows: pairs of complex numbers with form a 3-sphere, and their ratios cover the complex plane plus infinity, a 2-sphere. The Hopf map sends any such pair to its ratio.
Similarly, there are generalized Hopf fibrations
constructed using pairs of quaternions or octonions instead of complex numbers . Here, too, and are zero. Thus the long exact sequences again break into families of split short exact sequences, implying two families of relations.
The three fibrations have base space with , for . A fibration does exist for (), but not for () and beyond. Although generalizations of the relations to are often true, they sometimes fail; for example,
Thus there can be no fibration
the first non-trivial case of the Hopf invariant one problem, because such a fibration would imply that the failed relation is true.
Framed cobordism.
Homotopy groups of spheres are closely related to cobordism classes of manifolds.
In 1938 Lev Pontryagin established an isomorphism between the homotopy group and the group of cobordism classes of differentiable -submanifolds of which are "framed", i.e. have a trivialized normal bundle. Every map is homotopic to a differentiable map with formula_17 a framed -dimensional submanifold. For example, is the cobordism group of framed 0-dimensional submanifolds of , computed by the algebraic sum of their points, corresponding to the degree of maps formula_18. The projection of the Hopf fibration formula_19 represents a generator of which corresponds to the framed 1-dimensional submanifold of defined by the standard embedding formula_20 with a nonstandard trivialization of the normal 2-plane bundle. Until the advent of more sophisticated algebraic methods in the early 1950s (Serre) the Pontrjagin isomorphism was the main tool for computing the homotopy groups of spheres. In 1954 the Pontrjagin isomorphism was generalized by René Thom to an isomorphism expressing other groups of cobordism classes (e.g. of all manifolds) as homotopy groups of spaces and spectra. In more recent work the argument is usually reversed, with cobordism groups computed in terms of homotopy groups .
Finiteness and torsion.
In 1951, Jean-Pierre Serre showed that homotopy groups of spheres are all finite except for those of the form or (for positive ), when the group is the product of the infinite cyclic group with a finite abelian group . In particular the homotopy groups are determined by their -components for all primes . The 2-components are hardest to calculate, and in several ways behave differently from the -components for odd primes.
In the same paper, Serre found the first place that -torsion occurs in the homotopy groups of dimensional spheres, by showing that has no -torsion if , and has a unique subgroup of order if and . The case of 2-dimensional spheres is slightly different: the first -torsion occurs for . In the case of odd torsion there are more precise results; in this case there is a big difference between odd and even dimensional spheres. If is an odd prime and , then elements of the -component of have order at most . This is in some sense the best possible result, as these groups are known to have elements of this order for some values of . Furthermore, the stable range can be extended in this case: if is odd then the double suspension from to is an isomorphism of -components if , and an epimorphism if equality holds . The -torsion of the intermediate group can be strictly larger.
The results above about odd torsion only hold for odd-dimensional spheres: for even-dimensional spheres, the James fibration gives the torsion at odd primes in terms of that of odd-dimensional spheres,
(where means take the -component) . This exact sequence is similar to the ones coming from the Hopf fibration; the difference is that it works for all even-dimensional spheres, albeit at the expense of ignoring 2-torsion. Combining the results for odd and even dimensional spheres shows that much of the odd torsion of unstable homotopy groups is determined by the odd torsion of the stable homotopy groups.
For stable homotopy groups there are more precise results about -torsion. For example, if for a prime then the -primary component of the stable homotopy group vanishes unless is divisible by , in which case it is cyclic of order .
The J-homomorphism.
An important subgroup of , for , is the image of the J-homomorphism
, where denotes the special orthogonal group . In the stable range , the homotopy groups only depend on . This period 8 pattern is known as Bott periodicity, and it is reflected in the stable homotopy groups of spheres via the image of the -homomorphism which is:
This last case accounts for the elements of unusually large finite order in for such values of . For example, the stable groups have a cyclic subgroup of order 504, the denominator of .
The stable homotopy groups of spheres are the direct sum of the image of the -homomorphism, and the kernel of the Adams -invariant, a homomorphism from these groups to . Roughly speaking, the image of the -homomorphism is the subgroup of "well understood" or "easy" elements of the stable homotopy groups. These well understood elements account for most elements of the stable homotopy groups of spheres in small dimensions. The quotient of by the image of the -homomorphism is considered to be the "hard" part of the stable homotopy groups of spheres . (Adams also introduced certain order 2 elements of for , and these are also considered to be "well understood".) Tables of homotopy groups of spheres sometimes omit the "easy" part to save space.
Ring structure.
The direct sum
of the stable homotopy groups of spheres is a supercommutative graded ring, where multiplication is given by composition of representing maps, and any element of non-zero degree is nilpotent ; the nilpotence theorem on complex cobordism implies Nishida's theorem.
Example: If is the generator of (of order 2),
then is nonzero and generates , and is nonzero and 12 times a generator of , while is zero because the group is trivial.
If and and are elements of with and , there is a Toda bracket of these elements . The Toda bracket is not quite an element of a stable homotopy group, because it is only defined up to addition of products of certain other elements. Hiroshi Toda used the composition product and Toda brackets to label many of the elements of homotopy groups. There are also higher Toda brackets of several elements, defined when suitable lower Toda brackets vanish. This parallels the theory of Massey products in cohomology.
Every element of the stable homotopy groups of spheres can be expressed using composition products and higher Toda brackets in terms of certain well known elements, called Hopf elements .
Computational methods.
If is any finite simplicial complex with finite fundamental group, in particular if is a sphere of dimension at least 2, then its homotopy groups are all finitely generated abelian groups. To compute these groups, they are often factored into their -components for each prime , and calculating each of these -groups separately. The first few homotopy groups of spheres can be computed using ad hoc variations of the ideas above; beyond this point, most methods for computing homotopy groups of spheres are based on spectral sequences . This is usually done by constructing suitable fibrations and taking the associated long exact sequences of homotopy groups; spectral sequences are a systematic way of organizing the complicated information that this process generates.
The computation of the homotopy groups of has been reduced to a combinatorial group theory question. identify these homotopy groups as certain quotients of the Brunnian braid groups of . Under this correspondence, every nontrivial element in for may be represented by a Brunnian braid over that is not Brunnian over the disk . For example, the Hopf map corresponds to the Borromean rings.
Applications.
where is the cyclic subgroup represented by homotopy spheres that bound a parallelizable manifold, is the th stable homotopy group of spheres, and is the image of the -homomorphism. This is an isomorphism unless is of the form , in which case the image has index 1 or 2 .
Table of homotopy groups.
Tables of homotopy groups of spheres are most conveniently organized by showing .
The following table shows many of the groups . (These tables are based on the table of homotopy groups of spheres in .) The stable homotopy groups are highlighted in blue, the unstable ones in red. Each homotopy group is the product of the cyclic groups of the orders given in the table, using the following conventions:
Example: , which is denoted by in the table.
Table of stable homotopy groups.
The stable homotopy groups are the product of cyclic groups of the infinite or prime power orders
shown 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 , the -components in the range of the table are accounted for by the -homomorphism and are cyclic of order if divides and 0 otherwise . (The 2-components can be found in , and the 3- and 5-components in .) The mod 8 behavior of the table comes from Bott periodicity via the J-homomorphism, whose image is underlined.