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 mod 8 behavior of the table comes from Bott periodicity via the J-homomorphism, whose image is underlined. References Notes Sources . See also . . . . . . . . . . . . Also see the corrections in . . Pontrjagin, Lev, Smooth manifolds and their applications in homotopy theory American Mathematical Society Translations, Ser. 2, Vol. 11, pp. 1–114 (1959) . . . . . . . General algebraic topology references . . Historical papers . . . External links in MacTutor History of Mathematics archive. in MacTutor History of Mathematics archive. Homotopy theory Spheres