Homotopy groups of spheres: Difference between revisions

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.) The main complexity is in the 2-, 3-, and 5-components: Forfor {{math|''p'' > 5}}, the {{mvar|p}}-components in the range of the table are accounted for by the {{mvar|J}}-homomorphism and are cyclic of order {{mvar|p}} if {{math|2(''p'' &minus; 1)}} divides {{math|''k'' + 1}} and 0 otherwise.<ref>{{harvnb|Fuks|2001}}. The 2-components can be found in {{harvtxt|Isaksen|Wang|Xu|2020}}, and the 3- and 5-components in {{harvtxt|Ravenel|2003}}.</ref> The mod 8 behavior of the table comes from [[Bott periodicity]] via the [[J-homomorphism]], whose image is underlined.