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 Sn — generalizes the familiar circle (S1) and the ordinary sphere (S2). 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 πi(Sn) summarizes the different ways in which the i-dimensional sphere Si can be mapped continuously into the n-dimensional sphere Sn. 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 πi(Sn) falls into three regimes, depending on whether i is less than, equal to, or greater than n.
The question of computing the homotopy group πn+k(Sn) 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 πn+k(Sn) 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 < 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.
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.
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 (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.
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 (S1,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 S1↦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, πi(X) begins with the pointed i-sphere (Si,s), and otherwise follows the same procedure. The null homotopic class acts as the identity of the group addition, and for X equal to Sn (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 πi 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.
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 (Hatcher 2002). 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 π1(S1) 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.
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, πn(Sn) = ℤ (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 i < n, then πi(Sn) = 0. 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, π2(S1) = 0. This is because S1 has the real line as its universal cover which is contractible (it has the homotopy type of a point). In addition, because S2 is simply connected, by the lifting criterion, any map from S2 to S1 can be lifted to a map into the real line and the nullhomotopy descends to the downstairs space.
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 S3 to S2, now known as the Hopf fibration (Hopf 1931). This map generates the homotopy group π3(S2) = ℤ.
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 (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 (O'Connor & Robertson 1996).
Higher homotopy groups were first defined by Eduard Čech in 1932 (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 (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 (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 πn(Sn) and π4n−1(S2n). Others who worked in this area included José Ádem, Hiroshi Toda, Frank Adams and J. Peter May. The stable homotopy groups πn+k(Sn) are known for k up to 64, and, as of 2007, unknown for larger k (Hatcher 2002, Stable homotopy groups, pp. 385–393).
As noted already, when i is less than n, πi(Sn) = 0, the trivial group (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 Sn 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 πk(X), with k > 0, is isomorphic to the first nonzero homology group Hk(X). For the n-sphere, this immediately implies that for n ≥ 2, πn(Sn) = Hn(Sn) = ℤ.
The homology groups Hi(Sn), with i > 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 πi(Sn), for i > 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 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 ℤn), or direct products of such groups (written, for example, as ℤ24×ℤ3 or ). 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 πi(S0) of the 0-dimensional sphere are trivial for i > 0, because any base point preserving map from an i-sphere to a 0-sphere is a one-point mapping. Similarly, the homotopy groups πi(S1) of the 1-sphere are trivial for i > 1, because the universal covering space, ℝ, which has the same higher homotopy groups, is contractible.
Beyond these two rows, the higher homotopy groups (i > n) 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.
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 πn+k(Sn) to πn+k+1(Sn+1) is an isomorphism for n > k + 1. The groups πn+k(Sn) with n > k + 1 are called the stable homotopy groups of spheres, and are denoted πS
k: they are finite abelian groups for k ≠ 0, and have been computed in numerous cases, although the general pattern is still elusive. (Hatcher 2002, Stable homotopy groups, pp. 385–393). For n ≤ k+1, the groups are called the unstable homotopy groups of spheres.
The general theory of fiber bundles F→E→B shows that there is a long exact sequence of homotopy groups
For this specific bundle, each group homomorphism πi(S1)→πi(S3), induced by the inclusion S1→S3, maps all of πi(S1) to zero, since the lower-dimensional sphere S1 can be deformed to a point inside the higher-dimensional one S3. This corresponds to the vanishing of π1(S3). Thus the long exact sequence breaks into short exact sequences,
Since πi−1(S1) vanishes for i at least 3, the first row shows that πi(S2) and πi(S3) are isomorphic whenever i is at least 3, as observed above.
The Hopf fibration may be constructed as follows: pairs of complex numbers (z0,z1) with |z0|2 + |z1|2 = 1 form a 3-sphere, and their ratios z0⁄z1 cover the complex plane plus infinity, a 2-sphere. The Hopf map S3 → S2 sends any such pair to its ratio.
Similarly, there are generalized Hopf fibrations
constructed using pairs of quaternions or octonions instead of complex numbers (Hatcher 2002). Here, too, π3(S7) and π7(S15) 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 Sn with n = 2m, for m = 1, 2, 3. A fibration does exist for S1 (m = 0), but not for S16 (m = 4) and beyond. Although generalizations of the relations to S16 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.
Homotopy groups of spheres are closely related to cobordism classes of manifolds.
In 1938 Lev Pontryagin established an isomorphism between the homotopy group πn+k(Sn) and the group Ωframed
k(Sn+k) of cobordism classes of differentiable k-submanifolds of Sn+k which are "framed", i.e. have a trivialized normal bundle. Every map ƒ:Sn+k → Sn is homotopic to a differentiable map with a framed k-dimensional submanifold. For example, πn(Sn)=ℤ is the cobordism group of framed 0-dimensional submanifolds of Sn, computed by the algebraic sum of their points, corresponding to the degree of maps . The projection of the Hopf fibration represents a generator of π3(S2) = Ωframed
1(S3)=ℤ which corresponds to the framed 1-dimensional submanifold of S3 defined by the standard embedding 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 (Scorpan 2005).
In 1951, Jean-Pierre Serre showed that homotopy groups of spheres are all finite except for those of the form πn(Sn) or π4n−1(S2n) (for positive n), when the group is the product of the infinite cyclic group with a finite abelian group (Serre 1951). In particular the homotopy groups are determined by their p-components for all primes p. The 2-components are hardest to calculate, and in several ways behave differently from the p-components for odd primes.
In the same paper, Serre found the first place that p-torsion occurs in the homotopy groups of n dimensional spheres, by showing that πn+k(Sn) has no p-torsion if k < 2p − 3, and has a unique subgroup of order p if n ≥ 3 and k = 2p − 3. The case of 2-dimensional spheres is slightly different: the first p-torsion occurs for k = 2p − 3 + 1. 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 p is an odd prime and n = 2i + 1, then elements of the p-component of πn+k(Sn) have order at most pi (Cohen, Moore & Neisendorfer 1979). This is in some sense the best possible result, as these groups are known to have elements of this order for some values of k (Ravenel 2003, p. 4). Furthermore, the stable range can be extended in this case: if n is odd then the double suspension from πk(Sn) to πk+2(Sn+2) is an isomorphism of p-components if k < p(n + 1) − 3, and an epimorphism if equality holds (Serre 1952). The p-torsion of the intermediate group πk+1(Sn+1) 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 p in terms of that of odd-dimensional spheres,
(where (p) means take the p-component) (Ravenel 2003, p. 25). 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 p-torsion. For example, if k < 2p(p − 1) − 2 for a prime p then the p-primary component of the stable homotopy group πS
k vanishes unless k + 1 is divisible by 2(p − 1), in which case it is cyclic of order p (Fuks 2001) harv error: no target: CITEREFFuks2001 (help).
An important subgroup of πn+k(Sn), for k ≥ 2, is the image of the J-homomorphism J: πk(SO(n)) → πn+k(Sn), where SO(n) denotes the special orthogonal group (Adams 1966). In the stable range n ≥ k+2, the homotopy groups πk(SO(n)) only depend on k (mod 8). 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 J-homomorphism which is:
This last case accounts for the elements of unusually large finite order in πn+k(Sn) for such values of k. For example, the stable groups πn+11(Sn) have a cyclic subgroup of order 504, the denominator of B6⁄12 = 1⁄504.
The stable homotopy groups of spheres are the direct sum of the image of the J-homomorphism, and the kernel of the Adams e-invariant, a homomorphism from these groups to ℚ/ℤ. Roughly speaking, the image of the J-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 πS
n by the image of the J-homomorphism is considered to be the "hard" part of the stable homotopy groups of spheres (Adams 1966). (Adams also introduced certain order 2 elements μn of πS
n for n ≡ 1 or 2 (mod 8), and these are also considered to be "well understood".) Tables of homotopy groups of spheres sometimes omit the "easy" part im(J) to save space.
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 (Nishida 1973); the nilpotence theorem on complex cobordism implies Nishida's theorem.
Example: If η is the generator of πS
1 (of order 2), then η2 is nonzero and generates πS
2, and η3 is nonzero and 12 times a generator of πS
3, while η4 is zero because the group πS
4 is trivial.
If f and g and h are elements of πS
* with f g = 0 and g⋅h = 0, there is a Toda bracket 〈f,g,h〉 of these elements (Toda 1962). 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 (Cohen 1968).
If X is any finite simplicial complex with finite fundamental group, in particular if X 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 p-components for each prime p, and calculating each of these p-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 (Ravenel 2003). 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 S2 has been reduced to a combinatorial group theory question. Berrick et al. (2006) identify these homotopy groups as certain quotients of the Brunnian braid groups of S2. Under this correspondence, every nontrivial element in πn(S2) for n > 2 may be represented by a Brunnian braid over S2 that is not Brunnian over the disk D2. For example, the Hopf map S3 → S2 corresponds to the Borromean rings.
where bPn+1 is the cyclic subgroup represented by homotopy spheres that bound a parallelizable manifold, πS
n is the nth stable homotopy group of spheres, and J is the image of the J-homomorphism. This is an isomorphism unless n is of the form 2k−2, in which case the image has index 1 or 2 (Kervaire & Milnor 1963).
Tables of homotopy groups of spheres are most conveniently organized by showing πn+k(Sn).
The following table shows many of the groups πn+k(Sn). (These tables are based on the table of homotopy groups of spheres in Toda (1962).) 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: π19(S10) = π9+10(S10) = ℤ×ℤ2×ℤ2×ℤ2, which is denoted by ∞⋅23 in the table.
The stable homotopy groups πk 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 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 (Fuks 2001) harv error: no target: CITEREFFuks2001 (help). (The 2-components can be found in Kochman (1990), though there were some errors for k≥54 that were corrected by Kochman & Mahowald (1995), and the 3- and 5-components in Ravenel (2003).) The mod 8 behavior of the table comes from Bott periodicity via the J-homomorphism, whose image is underlined.