Homotopy groups of spheres#Computational methods

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.

• Implicit surface: {{math|x{{su|lh=1|b=0|p=2}} + x{{su|lh=1|b=1|p=2}} + x{{su|lh=1|b=2|p=2}} {{=}} 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, {{math|S²}}, for reasons given below. The same idea applies for any dimension {{mvar|n}}; the equation {{math|x{{su|lh=1|b=0|p=2}} + x{{su|lh=1|b=1|p=2}} + ⋯ + x{{su|lh=1|b=n|p=2}} {{=}} 1}} produces the n-sphere as a geometric object in ({{math|n + 1}})-dimensional space. For example, the 1-sphere {{math|S¹}} is a circle.{{sfn|Hatcher|2002|p=xii}}

• Disk with collapsed rim: written in topology as {{math|D²/S¹}}

: This construction moves from geometry to pure topology. The disk {{math|D²}} is the region contained by a circle, described by the inequality {{math|x{{su|lh=1|b=0|p=2}} + x{{su|lh=1|b=1|p=2}} ≤ 1}}, and its rim (or "boundary") is the circle {{math|S¹}}, described by the equality {{math|x{{su|lh=1|b=0|p=2}} + x{{su|lh=1|b=1|p=2}} {{=}} 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, {{math|Dⁿ/Sⁿ⁻¹}} produces {{math|Sⁿ}}. For example, {{math|D¹}} is a line segment, and the construction joins its ends to make a circle. An equivalent description is that the boundary of an {{mvar|n}}-dimensional disk is glued to a point, producing a CW complex.{{sfn|Hatcher|2002|loc=Example 0.3, p. 6}}

• Suspension of equator: written in topology as {{math|ΣS¹}}

: This construction, though simple, is of great theoretical importance. Take the circle {{math|S¹}} 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 {{mvar|n}}, the {{mvar|n}}-sphere {{math|x{{su|lh=1|b=0|p=2}} + x{{su|lh=1|b=1|p=2}} + ⋯ + x{{su|lh=1|b=n|p=2}} {{=}} 1}} has as equator the ({{math|n − 1}})-sphere {{math|x{{su|lh=1|b=0|p=2}} + x{{su|lh=1|b=1|p=2}} + ⋯ + x{{su|lh=1|b=n−1|p=2}} {{=}} 1}}, and the suspension {{math|ΣSⁿ⁻¹}} produces {{math|Sⁿ}}.{{sfn|Hatcher|2002|p=129}}

Some theory requires selecting a fixed point on the sphere, calling the pair {{math|(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.{{sfn|Hatcher|2002|p=28}} For spheres constructed as a repeated suspension, the point {{math|(1, 0, 0, ..., 0)}}, 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.

Image:Homotopy of pointed circle maps.png

Image:Homotopy group addition.svg

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.{{sfn|Hatcher|2002|p=3}} 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.{{sfn|Miranda|1995|pp=123–125}}

The first homotopy group, or fundamental group, {{math|π₁(X)}} of a (path connected) topological space {{mvar|X}} thus begins with continuous maps from a pointed circle {{math|(S¹,s)}} to the pointed space {{math|(X,x)}}, where maps from one pair to another map {{mvar|s}} into {{mvar|x}}. These maps (or equivalently, closed curves) are grouped together into equivalence classes based on homotopy (keeping the "base point" {{mvar|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 {{math|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.{{sfn|Hu|1959|p=[https://books.google.com/books?id=iVhMPU0X2G4C&pg=PA107 107]}}

More generally, the {{mvar|i}}-th homotopy group, {{math|π_i(X)}} begins with the pointed {{mvar|i}}-sphere {{math|(Sⁱ, s)}}, and otherwise follows the same procedure. The null homotopic class acts as the identity of the group addition, and for {{mvar|X}} equal to {{math|Sⁿ}} (for positive {{mvar|n}}) — the homotopy groups of spheres — the groups are abelian and finitely generated. If for some {{mvar|i}} all maps are null homotopic, then the group {{math|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 {{mvar|i}}-th homotopy groups are isomorphic for all {{mvar|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.{{cn|date=February 2022}}

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.

Image:Fundamental group of the circle.svg

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 {{math|π₁(S¹)}} is therefore an infinite cyclic group, and is isomorphic to the group {{math|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.{{sfn|Hatcher|2002|p=29}}

The identification (a group isomorphism) of the homotopy group with the integers is often written as an equality: thus {{math|π₁(S¹) {{=}} Z}}.See, e.g., {{harvnb|Homotopy type theory|2013}}, Section 8.1, "$\backslash pi\_1(S^1)$".

Image:Sphere wrapped round itself.png

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, {{math|Z}}.{{citation needed|date=July 2024}}

These two results generalize: for all {{math|n > 0}}, {{math|π_n(Sⁿ) {{=}} Z}} (see below).

Image:P1S2all.jpg

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 {{math|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.{{sfn|Hatcher|2002|p=348}}

This result generalizes to higher dimensions. All mappings from a lower-dimensional sphere into a sphere of higher dimension are similarly trivial: if {{math|i < n}}, then {{math|π_i(Sⁿ) {{=}} 0}}. This can be shown as a consequence of the cellular approximation theorem.{{sfn|Hatcher|2002|p=349}}

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, {{math|π₂(S¹) {{=}} 0}}. This is because {{math|S¹}} has the real line as its universal cover which is contractible (it has the homotopy type of a point). In addition, because {{math|S²}} is simply connected, by the lifting criterion,{{sfn|Hatcher|2002|p=61}} any map from {{math|S²}} to {{math|S¹}} can be lifted to a map into the real line and the nullhomotopy descends to the downstairs space (via composition).

Image:Hopf Fibration.png is a nontrivial mapping of the 3-sphere to the 2-sphere, and generates the third homotopy group of the 2-sphere. Each colored circle maps to the corresponding point on the 2-sphere shown bottom right.]]

The first nontrivial example with {{math|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 {{math|S³}} to {{math|S²}}, now known as the Hopf fibration.{{sfn|Hopf|1931}} This map generates the homotopy group {{math|π₃(S²) {{=}} Z}}.{{sfn|Walschap|2004|p=90}}

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.{{sfn|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.{{sfn|O'Connor|Robertson|1996}}

Higher homotopy groups were first defined by Eduard Čech in 1932.{{sfn|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.{{sfn|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.{{sfn|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 {{math|π_n(Sⁿ)}} and {{math|π_4n−1(S²ⁿ)}}. Others who worked in this area included José Adem, Hiroshi Toda, Frank Adams, J. Peter May, Mark Mahowald, Daniel Isaksen, Guozhen Wang, and Zhouli Xu. The stable homotopy groups {{math|π_n+k(Sⁿ)}} are known for {{mvar|k}} up to 90, and, as of 2023, unknown for larger {{mvar|k}}.{{sfn|Isaksen|Wang|Xu|2023}}

As noted already, when {{mvar|i}} is less than {{mvar|n}}, {{math|π_i(Sⁿ) {{=}} 0}}, the trivial group. The reason is that a continuous mapping from an {{mvar|i}}-sphere to an {{mvar|n}}-sphere with {{math|i < n}} can always be deformed so that it is not surjective. Consequently, its image is contained in {{math|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.{{sfn|Hatcher|2002|p=349}}

When {{math|i {{=}} n}}, {{math|π_n(Sⁿ) {{=}} Z}}, the infinite cyclic group, generated by the identity map from the {{mvar|n}}-sphere to itself. It follows from the definition of homotopy groups that the identity map and its multiples are elements of {{math|π_n(Sⁿ)}}. That these are the only elements can be shown using the Freudenthal suspension theorem, which relates the homotopy groups of a space and its suspension. In the case of spheres, the suspension of an {{mvar|n}}-sphere is an {{math|(n+1)}}-sphere, and the suspension theorem states that there is a group homomorphism {{math|π_i(Sⁿ) → π_i+1(Sⁿ⁺¹)}} which is an isomorphism for all {{math|i < 2n-1}} and is surjective for {{math|i {{=}} 2n-1}}. This implies that there is a sequence of group homomorphisms

:$\backslash pi\_1(S^1)\; \backslash to\; \backslash pi\_2(S^2)\; \backslash to\; \backslash pi\_3(S^3)\; \backslash to\; \backslash cdots$

in which the first homomorphism is a surjection and the rest are isomorphisms. As noted already, {{math|π₁(S¹) {{=}} Z}}, and {{math|π₂(S²)}} contains a copy of {{math|Z}} generated by the identity map, so the fact that there is a surjective homomorphism from {{math|π₁(S¹)}} to {{math|π₂(S²)}} implies that {{math|π₂(S²) {{=}} Z}}. The rest of the homomorphisms in the sequence are isomorphisms, so {{math|π_n(Sⁿ) {{=}} Z}} for all {{mvar|n}}.{{sfn|Hatcher|2002|p=361}}

The homology groups {{math|H_i(Sⁿ)}}, with {{math|i > n}}, are all trivial. It therefore came as a great surprise historically that the corresponding homotopy groups are not trivial in general.{{cn|date=January 2025}} This is the case that is of real importance: the higher homotopy groups {{math|π_i(Sⁿ)}}, for {{math|i > n}}, are surprisingly complex and difficult to compute, and the effort to compute them has generated a significant amount of new mathematics.{{cn|date=February 2022}}

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 a) the trivial group 0, the infinite cyclic group {{math|Z}}, b) the finite cyclic groups of order {{mvar|n}} (written as {{math|Z_n}}), or c) the direct products of such groups (written, for example, as {{math|Z₂₄×Z₃}} or {{math|1=Z{{su|lh=1|b=2|p=2}} = Z₂×Z₂}}). Extended tables of homotopy groups of spheres are given at the end of the article.

The first row of this table is straightforward. The homotopy groups {{math|π_i(S¹)}} of the 1-sphere are trivial for {{math|i > 1}}, because the universal covering space, $\backslash mathbb\{R\}$, which has the same higher homotopy groups, is contractible.{{sfn|Hatcher|2002|p=342}}

Beyond the first row, the higher homotopy groups ({{math|i > n}}) appear to be chaotic, but in fact there are many patterns, some obvious and some very subtle.

• The groups below the jagged black line are constant along the diagonals (as indicated by the red, green and blue coloring).
• Most of the groups are finite. The only infinite groups are either on the main diagonal or immediately above the jagged line (highlighted in yellow).
• The second and third rows of the table are the same starting in the third column (i.e., {{math|π_i(S²) {{=}} π_i(S³)}} for {{math| i ≥ 3}}). This isomorphism is induced by the Hopf fibration {{math| S³ → S²}}.
• For {{math|1=n = 2, 3, 4, 5}} and {{math|i ≥ n}} the homotopy groups {{math|π_i(Sⁿ)}} do not vanish. However, {{math|1=π_n+4(Sⁿ) = 0}} for {{math|n ≥ 6}}.

These patterns follow from many different theoretical results.{{cn|date=February 2022}}

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 {{math|π_n+k(Sⁿ)}} to {{math|π_n+k+1(Sⁿ⁺¹)}} is an isomorphism for {{math|n > k + 1}}. The groups {{math|π_n+k(Sⁿ)}} with {{math|n > k + 1}} are called the stable homotopy groups of spheres, and are denoted {{math|π{{su|lh=1|b=k|p=S}}}}: they are finite abelian groups for {{math|k ≠ 0}}, and have been computed in numerous cases, although the general pattern is still elusive.{{sfn|Hatcher|2002|loc=Stable homotopy groups, pp. 385–393}} For {{math|n ≤ k+1}}, the groups are called the unstable homotopy groups of spheres.{{cn|date=February 2022}}

The classical Hopf fibration is a fiber bundle:

:$S^1\backslash hookrightarrow\; S^3\backslash rightarrow\; S^2.$

The general theory of fiber bundles {{math|F → E → B}} shows that there is a long exact sequence of homotopy groups

:$\backslash cdots\; \backslash to\; \backslash pi\_i(F)\; \backslash to\; \backslash pi\_i(E)\; \backslash to\; \backslash pi\_i(B)\; \backslash to\; \backslash pi\_\{i-1\}(F)\; \backslash to\; \backslash cdots.$

For this specific bundle, each group homomorphism {{math|π_i(S¹) → π_i(S³)}}, induced by the inclusion {{math|S¹ → S³}}, maps all of {{math|π_i(S¹)}} to zero, since the lower-dimensional sphere {{math|S¹}} can be deformed to a point inside the higher-dimensional one {{math|S³}}. This corresponds to the vanishing of {{math|π₁(S³)}}. Thus the long exact sequence breaks into short exact sequences,

:$0\backslash rightarrow\; \backslash pi\_i(S^3)\backslash rightarrow\; \backslash pi\_i(S^2)\backslash rightarrow\; \backslash pi\_\{i-1\}(S^1)\backslash rightarrow\; 0\; .$

Since {{math|Sⁿ⁺¹}} is a suspension of {{math|Sⁿ}}, these sequences are split by the suspension homomorphism {{math|π_i−1(S¹) → π_i(S²)}}, giving isomorphisms

:$\backslash pi\_i(S^2)=\; \backslash pi\_i(S^3)\backslash oplus\; \backslash pi\_\{i-1\}(S^1)\; .$

Since {{math|π_i−1(S¹)}} vanishes for {{mvar|i}} at least 3, the first row shows that {{math|π_i(S²)}} and {{math|π_i(S³)}} are isomorphic whenever {{mvar|i}} is at least 3, as observed above.

The Hopf fibration may be constructed as follows: pairs of complex numbers {{math|(z₀,z₁)}} with {{math|{{abs|z₀}}² + {{abs|z₁}}² {{=}} 1}} form a 3-sphere, and their ratios {{math|{{sfrac|z₀|z₁}}}} cover the complex plane plus infinity, a 2-sphere. The Hopf map {{math|S³ → S²}} sends any such pair to its ratio.{{cn|date=February 2022}}

Similarly (in addition to the Hopf fibration $S^0\backslash hookrightarrow\; S^1\backslash rightarrow\; S^1$, where the bundle projection is a double covering), there are generalized Hopf fibrations

:$S^3\backslash hookrightarrow\; S^7\backslash rightarrow\; S^4$

:$S^7\backslash hookrightarrow\; S^\{15\}\backslash rightarrow\; S^8$

constructed using pairs of quaternions or octonions instead of complex numbers.{{sfn|Hatcher|2002}} Here, too, {{math|π₃(S⁷)}} and {{math|π₇(S¹⁵)}} are zero. Thus the long exact sequences again break into families of split short exact sequences, implying two families of relations.

:$\backslash pi\_i(S^4)=\; \backslash pi\_i(S^7)\backslash oplus\; \backslash pi\_\{i-1\}(S^3)\; ,$

:$\backslash pi\_i(S^8)=\; \backslash pi\_i(S^\{15\})\backslash oplus\; \backslash pi\_\{i-1\}(S^7)\; .$

The three fibrations have base space {{math|Sⁿ}} with {{math|n {{=}} 2^m}}, for {{math|m {{=}} 1, 2, 3}}. A fibration does exist for {{math|S¹}} ({{math|m {{=}} 0}}) as mentioned above, but not for {{math|S¹⁶}} ({{math|m {{=}} 4}}) and beyond. Although generalizations of the relations to {{math|S¹⁶}} are often true, they sometimes fail; for example,

:$\backslash pi\_\{30\}(S^\{16\})\backslash neq\; \backslash pi\_\{30\}(S^\{31\})\backslash oplus\; \backslash pi\_\{29\}(S^\{15\})\; .$

Thus there can be no fibration

:$S^\{15\}\backslash hookrightarrow\; S^\{31\}\backslash rightarrow\; S^\{16\}\; ,$

the first non-trivial case of the Hopf invariant one problem, because such a fibration would imply that the failed relation is true.{{cn|date=February 2022}}

Homotopy groups of spheres are closely related to cobordism classes of manifolds.

In 1938 Lev Pontryagin established an isomorphism between the homotopy group {{math|π_n+k(Sⁿ)}} and the group {{math|Ω{{su|lh=1|b=k|p=framed}}(S^n+k)}} of cobordism classes of differentiable {{mvar|k}}-submanifolds of {{math|S^n+k}} which are "framed", i.e. have a trivialized normal bundle. Every map {{math|f : S^n+k → Sⁿ}} is homotopic to a differentiable map with {{math|1=M^k = f⁻¹(1, 0, ..., 0) ⊂ S^n+k}} a framed {{mvar|k}}-dimensional submanifold. For example, {{math|π_n(Sⁿ) {{=}} Z}} is the cobordism group of framed 0-dimensional submanifolds of {{math|Sⁿ}}, computed by the algebraic sum of their points, corresponding to the degree of maps {{math|f : Sⁿ → Sⁿ}}. The projection of the Hopf fibration {{math|S³ → S²}} represents a generator of {{math|π₃(S²) {{=}} Ω{{su|lh=1|b=1|p=framed}}(S³) {{=}} Z}} which corresponds to the framed 1-dimensional submanifold of {{math|S³}} defined by the standard embedding {{math|S¹ ⊂ S³}} 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.{{sfn|Scorpan|2005}}

In 1951, Jean-Pierre Serre showed that homotopy groups of spheres are all finite except for those of the form {{math|π_n(Sⁿ)}} or {{math|π_4n−1(S²ⁿ)}} (for positive {{mvar|n}}), when the group is the product of the infinite cyclic group with a finite abelian group.{{sfn|Serre|1951}} In particular the homotopy groups are determined by their {{mvar|p}}-components for all primes {{mvar|p}}. The 2-components are hardest to calculate, and in several ways behave differently from the {{mvar|p}}-components for odd primes.{{cn|date=February 2022}}

In the same paper, Serre found the first place that {{mvar|p}}-torsion occurs in the homotopy groups of {{mvar|n}} dimensional spheres, by showing that {{math|π_n+k(Sⁿ)}} has no {{mvar|p}}-torsion if {{math|k < 2p − 3}}, and has a unique subgroup of order {{mvar|p}} if {{math|n ≥ 3}} and {{math|k {{=}} 2p − 3}}. The case of 2-dimensional spheres is slightly different: the first {{mvar|p}}-torsion occurs for {{math|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 {{mvar|p}} is an odd prime and {{math|n {{=}} 2i + 1}}, then elements of the {{mvar|p}}-component of {{math|π_n+k(Sⁿ)}} have order at most {{math|pⁱ}}.{{sfn|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 {{mvar|k}}.{{sfn|Ravenel|2003|p=4}} Furthermore, the stable range can be extended in this case: if {{mvar|n}} is odd then the double suspension from {{math|π_k(Sⁿ)}} to {{math|π_k+2(Sⁿ⁺²)}} is an isomorphism of {{mvar|p}}-components if {{math|k < p(n + 1) − 3}}, and an epimorphism if equality holds.{{sfn|Serre|1952}} The {{mvar|p}}-torsion of the intermediate group {{math|π_k+1(Sⁿ⁺¹)}} can be strictly larger.{{cn|date=February 2022}}

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 {{mvar|p}} in terms of that of odd-dimensional spheres,

:$\backslash pi\_\{2m+k\}(S^\{2m\})(p)\; =\; \backslash pi\_\{2m+k-1\}(S^\{2m-1\})(p)\backslash oplus\; \backslash pi\_\{2m+k\}(S^\{4m-1\})(p)$

(where {{math|(p)}} means take the {{mvar|p}}-component).{{sfn|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.{{cn|date=February 2022}}

For stable homotopy groups there are more precise results about {{mvar|p}}-torsion. For example, if {{math|k < 2p(p − 1) − 2}} for a prime {{mvar|p}} then the {{mvar|p}}-primary component of the stable homotopy group {{math|π{{su|lh=1|b=k|p=S}}}} vanishes unless {{math|k + 1}} is divisible by {{math|2(p − 1)}}, in which case it is cyclic of order {{mvar|p}}.{{sfn|Fuks|2001}}

{{main|J-homomorphism}}

An important subgroup of {{math|π_n+k(Sⁿ)}}, for {{math|k ≥ 2}}, is the image of the J-homomorphism

{{math|J : π_k(SO(n)) → π_n+k(Sⁿ)}}, where {{math|SO(n)}} denotes the special orthogonal group.{{sfn|Adams|1966}} In the stable range {{math|n ≥ k + 2}}, the homotopy groups {{math|π_k(SO(n))}} only depend on {{math|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 {{mvar|J}}-homomorphism which is:

• a cyclic group of order 2 if {{mvar|k}} is congruent to 0 or 1 modulo 8;
• trivial if {{mvar|k}} is congruent to 2, 4, 5, or 6 modulo 8; and
• a cyclic group of order equal to the denominator of {{math|{{sfrac|B_2m|4m}}}}, where {{math|B_2m}} is a Bernoulli number, if {{math|k {{=}} 4m − 1 ≡ 3 (mod 4)}}.

This last case accounts for the elements of unusually large finite order in {{math|π_n+k(Sⁿ)}} for such values of {{mvar|k}}. For example, the stable groups {{math|π_n+11(Sⁿ)}} have a cyclic subgroup of order 504, the denominator of {{math|{{sfrac|B₆|12}} {{=}} {{sfrac|1|504}}}}.{{cn|date=February 2022}}

The stable homotopy groups of spheres are the direct sum of the image of the {{mvar|J}}-homomorphism, and the kernel of the Adams {{mvar|e}}-invariant, a homomorphism from these groups to {{math|$\backslash mathbb\{Q\}\; /\; \backslash mathbb\{Z\}$}}. Roughly speaking, the image of the {{mvar|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 {{math|π{{su|lh=1|b=n|p=S}}}} by the image of the {{mvar|J}}-homomorphism is considered to be the "hard" part of the stable homotopy groups of spheres {{harv|Adams|1966}}. (Adams also introduced certain order 2 elements {{math|μ_n}} of {{math|π{{su|lh=1|b=n|p=S}}}} for {{math|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 {{math|im(J)}} to save space.{{cn|date=February 2022}}

The direct sum

:$\backslash pi\_\{\backslash ast\}^S=\backslash bigoplus\_\{k\backslash ge\; 0\}\backslash pi\_k^S$

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;{{sfn|Nishida|1973}} the nilpotence theorem on complex cobordism implies Nishida's theorem.{{cn|date=February 2022}}

Example: If {{mvar|η}} is the generator of {{math|π{{su|lh=1|b=1|p=S}}}} (of order 2),

then {{math|η²}} is nonzero and generates {{math|π{{su|lh=1|b=2|p=S}}}}, and {{math|η³}} is nonzero and 12 times a generator of {{math|π{{su|lh=1|b=3|p=S}}}}, while {{math|η⁴}} is zero because the group {{math|π{{su|lh=1|b=4|p=S}}}} is trivial.{{cn|date=February 2022}}

If {{mvar|f}} and {{mvar|g}} and {{mvar|h}} are elements of {{math|π{{su|lh=1|b=*|p=S}}}} with {{math|f g {{=}} 0}} and {{math|g⋅h {{=}} 0}}, there is a Toda bracket {{math|{{angle bracket|f, g, h}}}} of these elements.{{sfn|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.{{cn|date=February 2022}}

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.{{sfn|Cohen|1968}}

If {{mvar|X}} is any finite simplicial complex with finite fundamental group, in particular if {{mvar|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 component (group theory) for each prime {{mvar|p}}, and calculating each of these p-group 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.{{sfn|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.{{cn|date=February 2022}}

• "The method of killing homotopy groups", due to {{harvs|txt=yes|last=Cartan|last2=Serre|year1=1952a|year2=1952b}} involves repeatedly using the Hurewicz theorem to compute the first non-trivial homotopy group and then killing (eliminating) it with a fibration involving an Eilenberg–MacLane space. In principle this gives an effective algorithm for computing all homotopy groups of any finite simply connected simplicial complex, but in practice it is too cumbersome to use for computing anything other than the first few nontrivial homotopy groups as the simplicial complex becomes much more complicated every time one kills a homotopy group.
• The Serre spectral sequence was used by Serre to prove some of the results mentioned previously. He used the fact that taking the loop space of a well behaved space shifts all the homotopy groups down by 1, so the {{mvar|n}}th homotopy group of a space {{mvar|X}} is the first homotopy group of its ({{math|n−1}})-fold repeated loop space, which is equal to the first homology group of the ({{math|n−1}})-fold loop space by the Hurewicz theorem. This reduces the calculation of homotopy groups of {{mvar|X}} to the calculation of homology groups of its repeated loop spaces. The Serre spectral sequence relates the homology of a space to that of its loop space, so can sometimes be used to calculate the homology of loop spaces. The Serre spectral sequence tends to have many non-zero differentials, which are hard to control, and too many ambiguities appear for higher homotopy groups. Consequently, it has been superseded by more powerful spectral sequences with fewer non-zero differentials, which give more information.{{cn|date=February 2022}}
• The EHP spectral sequence can be used to compute many homotopy groups of spheres; it is based on some fibrations used by Toda in his calculations of homotopy groups.{{sfn|Mahowald|2001}}{{sfn|Toda|1962}}
• The classical Adams spectral sequence has {{math|E₂}} term given by the Ext groups {{math|Ext{{su|lh=1|b=A(p)|p=∗,∗}}(Z_p, Z_p)}} over the mod {{mvar|p}} Steenrod algebra {{math|A(p)}}, and converges to something closely related to the {{mvar|p}}-component of the stable homotopy groups. The initial terms of the Adams spectral sequence are themselves quite hard to compute: this is sometimes done using an auxiliary spectral sequence called the May spectral sequence.{{sfn|Ravenel|2003|pp=67–74}}
• At the odd primes, the Adams–Novikov spectral sequence is a more powerful version of the Adams spectral sequence replacing ordinary cohomology mod {{mvar|p}} with a generalized cohomology theory, such as complex cobordism or, more usually, a piece of it called Brown–Peterson cohomology. The initial term is again quite hard to calculate; to do this one can use the chromatic spectral sequence.{{sfn|Ravenel|2003|loc=Chapter 5}}

Image:BorromeanRings.svg

• A variation of this last approach uses a backwards version of the Adams–Novikov spectral sequence for Brown–Peterson cohomology: the limit is known, and the initial terms involve unknown stable homotopy groups of spheres that one is trying to find.{{sfn|Kochman|1990}}

• The motivic Adams spectral sequence converges to the motivic stable homotopy groups of spheres. By comparing the motivic one over the complex numbers with the classical one, Isaksen gives rigorous proof of computations up to the 59-stem. In particular, Isaksen computes the Coker J of the 56-stem is 0, and therefore by the work of Kervaire-Milnor, the sphere {{math|S⁵⁶}} has a unique smooth structure.{{sfn|Isaksen|2019}}

• The Kahn–Priddy map induces a map of Adams spectral sequences from the suspension spectrum of infinite real projective space to the sphere spectrum. It is surjective on the Adams {{math|E₂}} page on positive stems. Wang and Xu develops a method using the Kahn–Priddy map to deduce Adams differentials for the sphere spectrum inductively. They give detailed argument for several Adams differentials and compute the 60 and 61-stem. A geometric corollary of their result is the sphere {{math|S⁶¹}} has a unique smooth structure, and it is the last odd dimensional one – the only ones are {{math|S¹}}, {{math|S³}}, {{math|S⁵}}, and {{math|S⁶¹}}.{{sfn|Wang|Xu|2017}}

• The motivic cofiber of {{math|τ}} method is so far the most efficient method at the prime 2. The class {{math|τ}} is a map between motivic spheres. The Gheorghe–Wang–Xu theorem identifies the motivic Adams spectral sequence for the cofiber of {{math|τ}} as the algebraic Novikov spectral sequence for {{math|BP_*}}, which allows one to deduce motivic Adams differentials for the cofiber of {{math|τ}} from purely algebraic data. One can then pullback these motivic Adams differentials to the motivic sphere, and then use the Betti realization functor to push forward them to the classical sphere.{{sfn|Gheorghe|Wang|Xu|2021}} Using this method, {{harvtxt|Isaksen|Wang|Xu|2023}} computes up to the 90-stem.{{sfn|Isaksen|Wang|Xu|2023}}

The computation of the homotopy groups of {{math|S²}} has been reduced to a combinatorial group theory question. {{harvtxt|Berrick|Cohen|Wong|Wu|2006}} identify these homotopy groups as certain quotients of the Brunnian braid groups of {{math|S²}}. Under this correspondence, every nontrivial element in {{math|π_n(S²)}} for {{math|n > 2}} may be represented by a Brunnian braid over {{math|S²}} that is not Brunnian over the disk {{math|D²}}. For example, the Hopf map {{math|S³ → S²}} corresponds to the Borromean rings.{{sfn|Berrick|Cohen|Wong|Wu|2006}}

• The winding number (corresponding to an integer of {{math|π₁(S¹) {{=}} Z)}} can be used to prove the fundamental theorem of algebra, which states that every non-constant complex polynomial has a zero.{{sfn|Fine|Rosenberger|1997}}
• The fact that {{math|π_n−1(Sⁿ⁻¹) {{=}} Z}} implies the Brouwer fixed point theorem that every continuous map from the {{mvar|n}}-dimensional ball to itself has a fixed point.{{sfn|Hatcher|2002|p=32}}
• The stable homotopy groups of spheres are important in singularity theory, which studies the structure of singular points of smooth maps or algebraic varieties. Such singularities arise as critical points of smooth maps from {{math|$\backslash mathbb\{R\}$^m}} to {{math|$\backslash mathbb\{R\}$ⁿ}}. The geometry near a critical point of such a map can be described by an element of {{math|π_m−1(Sⁿ⁻¹)}}, by considering the way in which a small {{math|m − 1}} sphere around the critical point maps into a topological {{math|n − 1}} sphere around the critical value.{{cn|date=February 2022}}
• The fact that the third stable homotopy group of spheres is cyclic of order 24, first proved by Vladimir Rokhlin, implies Rokhlin's theorem that the signature of a compact smooth spin 4-manifold is divisible by 16.{{sfn|Scorpan|2005}}
• Stable homotopy groups of spheres are used to describe the group {{math|Θ_n}} of h-cobordism classes of oriented homotopy {{mvar|n}}-spheres (for {{math|n ≠ 4}}, this is the group of smooth structures on {{mvar|n}}-spheres, up to orientation-preserving diffeomorphism; the non-trivial elements of this group are represented by exotic spheres). More precisely, there is an injective map

::$\backslash Theta\_n\; /\; bP\_\{n+1\}\; \backslash to\; \backslash pi\_n^S\; /\; J\; ,$

:where {{math|bP_n+1}} is the cyclic subgroup represented by homotopy spheres that bound a parallelizable manifold, {{math|π{{su|lh=1|b=n|p=S}}}} is the {{mvar|n}}th stable homotopy group of spheres, and {{mvar|J}} is the image of the J homomorphism. This is an isomorphism unless {{mvar|n}} is of the form {{math|2^k − 2}}, in which case the image has index 1 or 2.{{sfn|Kervaire|Milnor|1963}}

• The groups {{math|Θ_n}} above, and therefore the stable homotopy groups of spheres, are used in the classification of possible smooth structures on a topological or piecewise linear manifold.{{sfn|Scorpan|2005}}
• The Kervaire invariant problem, about the existence of manifolds of Kervaire invariant 1 in dimensions {{math|2^k − 2}} can be reduced to a question about stable homotopy groups of spheres. For example, knowledge of stable homotopy groups of degree up to 48 has been used to settle the Kervaire invariant problem in dimension {{math|2⁶ − 2 {{=}} 62}}. (This was the smallest value of {{mvar|k}} for which the question was open at the time.){{sfn|Barratt|Jones|Mahowald|1984}}
• The Barratt–Priddy theorem says that the stable homotopy groups of the spheres can be expressed in terms of the plus construction applied to the classifying space of the symmetric group, leading to an identification of K-theory of the field with one element with stable homotopy groups.{{sfn|Deitmar|2006}}

Tables of homotopy groups of spheres are most conveniently organized by showing {{math|π_n+k(Sⁿ)}}.

The following table shows many of the groups {{math|π_n+k(Sⁿ)}}. 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:These tables are based on the [http://www.math.nus.edu.sg/%7Ematwujie/homotopy_groups_sphere.html table of homotopy groups of spheres] in {{harvtxt|Toda|1962}}.

• The entry "⋅" denotes the trivial group.
• Where the entry is an integer, {{mvar|m}}, the homotopy group is the cyclic group of that order (generally written {{math|Z_m}}).
• Where the entry is ∞, the homotopy group is the infinite cyclic group, {{math|Z}}.
• Where entry is a product, the homotopy group is the cartesian product (equivalently, direct sum) of the cyclic groups of those orders. Powers indicate repeated products. (Note that when {{mvar|a}} and {{mvar|b}} have no common factor, {{math|Z_a×Z_b}} is isomorphic to {{math|Z_ab}}.)

Example: {{math|π₁₉(S¹⁰) {{=}} π₉₊₁₀(S¹⁰) {{=}} Z×Z₂×Z₂×Z₂}}, which is denoted by {{math|∞⋅2³}} in the table.

The stable homotopy groups {{math|π{{su|lh=1|b=k|p=S}}}} are the products 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.) For {{math|p > 5}}, the part of the {{mvar|p}}-component that is accounted for by the {{mvar|J}}-homomorphism is cyclic of order {{mvar|p}} if {{math|2(p − 1)}} divides {{math|k + 1}} and 0 otherwise.{{harvnb|Fuks|2001}}. The 2-components can be found in {{harvtxt|Isaksen|Wang|Xu|2023}}, 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.

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Category:Homotopy theory

Category:Spheres