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In mathematics, especially in the area of topology known as algebraic topology, the induced homomorphism is a group homomorphism related to the study of the fundamental group. ==Definition== Let ''X'' and ''Y'' be topological spaces; let ''x''0 be a point of ''X'' and let ''y''0 be a point of ''Y''. If ''h'' is a continuous map from ''X'' to ''Y'' such that . Define a map ''h''∗ from to by composing a loop in with ''h'' to get a loop in . Then ''h''∗ is a homomorphism between fundamental groups known as the homomorphism induced by ''h''. * If ''f'' is a loop in , then ''h''∗(''f'') is a loop in . It should be noted that ''h''∗(''f'') is a continuous map from to ''Y'', and and . * ''h'' is indeed a homomorphism. To avoid repetition, whenever we call ''f'' and ''g'' loops, they will be known as loops based at ''x''0. Let ''f'' and ''g'' be two loops, • be the group operation on and + be the group operation on , :''h''∗(''f'' • ''g'') = ''h''∗(''f''(2''t'')) for ''t'' in (1/2 ) = (''h''∗(''f'')) + (''h''∗(''g'')) :''h''∗(''f'' • ''g'') = ''h''∗(''g''(2''t'' − 1)) for ''t'' in (1 ) = (''h''∗(''f'')) + (''h''∗(''g'')) so that ''h''∗ is indeed a homomorphism. * Checking ''h''∗ is a function (i.e. every loop in gets mapped onto a unique loop in follows from the fact that if ''f'' and ''g'' are loops in that are homotopic via the homotopy ''H'', then ''h''∗(''f'') and ''h''∗(''g'') are homotopic via the homotopy ''h''∗''H''. 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「Induced homomorphism (fundamental group)」の詳細全文を読む スポンサード リンク
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