|
In algebraic topology, the Hopf construction constructs a map from the join ''X'' *''Y'' of two spaces ''X'' and ''Y'' to the suspension ''SZ'' of a space ''Z'' out of a map from ''X''×''Y'' to ''Z''. It was introduced by in the case when ''X'' and ''Y'' are spheres. used it to define the J-homomorphism. ==Construction== The Hopf construction can be obtained as the composition of a map :''X'' *''Y'' → ''S''(''X''×''Y'') and the suspension :''S''(''X''×''Y'') → ''S''(''Z'') of the map from ''X''×''Y'' to ''Z''. The map from ''X'' *''Y'' to ''S''(''X''×''Y'') can be obtained by regarding both sides as a quotient of ''X''×''Y''×''I'' where ''I'' is the unit interval. For ''X'' *''Y'' one identifies (''x'',''y'',0) with (''z'',''y'',0) and (''x'',''y'',1) with (''x'',''z'',1), while for ''S''(''X''×''Y'') one contracts all points of the form (''x'',''y'',0) to a point and also contracts all points of the form (''x'',''y'',1) to a point. So the map from ''X''×''Y''×''I'' to ''S''(''X''×''Y'') factors through ''X'' *''Y''. 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「Hopf construction」の詳細全文を読む スポンサード リンク
|