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In mathematics, the Toda bracket is an operation on homotopy classes of maps, in particular on homotopy groups of spheres, named after Hiroshi Toda who defined them and used them to compute homotopy groups of spheres in . == Definition == See or for more information. Suppose that : is a sequence of maps between spaces, such that ''gf'' and ''hg'' are both nullhomotopic. Given a space ''A'', let ''CA'' denote the cone of ''A''. Then we get a non-unique map from ''CW'' to ''Y'' from a homotopy from ''gf'' to a trivial map, which when composed with ''h'' gives a map from ''CW'' to ''Z''. Similarly we get a non-unique map from ''CX'' to ''Z'' from a homotopy from ''hg'' to a trivial map, which when composed with ''Cf'', the cone of the map ''f'', gives another map from ''CW'' to ''Z''. By joining together these two cones on ''W'' and the maps from them to ''Z'', we get a map 〈''f'', ''g'', ''h''〉 in the group (''Z'' ) of homotopy classes of maps from the suspension SW to ''Z'', called the Toda bracket of ''f'', ''g'', and ''h''. It is not uniquely defined up to homotopy, because there was some choice in choosing the maps from the cones. Changing these maps changes the Toda bracket by adding elements of ''h''() and ()''f''. 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. 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「Toda bracket」の詳細全文を読む スポンサード リンク
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