Editing Homotopy groups of spheres (section)

==Computational methods==
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 group]]s. To compute these groups, they are often factored into their [[component (group theory)|{{mvar|p}}-components]] for each [[prime number|prime]] {{mvar|p}}, and calculating each of these [[p-group|{{mvar|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 sequence]]s.{{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|''E''<sub>2</sub>}} term given by the [[Ext group]]s {{math|Ext{{su|lh=1|b=''A''(''p'')|p=∗,∗}}(Z<sub>''p''</sub>, Z<sub>''p''</sub>)}} 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|150px|thumb|Borromean rings]]
*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|''S''<sup>56</sup>}} 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|''E''<sub>2</sub>}} 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|''S''<sup>61</sup>}} has a unique smooth structure, and it is the last odd dimensional one – the only ones are {{math|''S''<sup>1</sup>}}, {{math|''S''<sup>3</sup>}}, {{math|''S''<sup>5</sup>}}, and {{math|''S''<sup>61</sup>}}.{{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''<sub>*</sub>}}, 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|''S''<sup>2</sup>}} has been reduced to a [[Geometric group theory|combinatorial group theory]] question. {{harvtxt|Berrick|Cohen|Wong|Wu|2006}} identify these homotopy groups as certain quotients of the [[Brunnian link|Brunnian]] [[braid group]]s of {{math|''S''<sup>2</sup>}}. Under this correspondence, every nontrivial element in {{math|π<sub>''n''</sub>(''S''<sup>2</sup>)}} for {{math|''n'' > 2}} may be represented by a Brunnian [[braid]] over {{math|''S''<sup>2</sup>}} that is not Brunnian over the disk {{math|''D''<sup>2</sup>}}. For example, the Hopf map {{math|''S''<sup>3</sup> → ''S''<sup>2</sup>}} corresponds to the [[Borromean rings]].{{sfn|Berrick|Cohen|Wong|Wu|2006}}