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In mathematics, the ''J''-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres. It was defined by , extending a construction of . ==Definition== Whitehead's original homomorphism is defined geometrically, and gives a homomorphism : of abelian groups for integers ''q'', and ''r'' ≥ 2. (Hopf defined this for the special case ''q''=''r''+1.) The ''J''-homomorphism can be defined as follows. An element of the special orthogonal group SO(''q'') can be regarded as a map : and the homotopy group π''r''(SO(''q'')) consists of homotopy-equivalence classes of maps from the ''r''-sphere to SO(''q''). Thus an element of π''r''(SO(''q'')) can be represented by a map : Applying the Hopf construction to this gives a map : in π''r''+''q''(''S''''q''), which Whitehead defined as the image of the element of π''r''(SO(''q'')) under the J-homomorphism. Taking a limit as ''q'' tends to infinity gives the stable ''J''-homomorphism in stable homotopy theory: : where SO is the infinite special orthogonal group, and the right-hand side is the ''r''-th stable stem of the stable homotopy groups of spheres. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「J-homomorphism」の詳細全文を読む スポンサード リンク
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