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In homotopy theory (a branch of mathematics), the Whitehead theorem states that if a continuous mapping ''f'' between topological spaces ''X'' and ''Y'' induces isomorphisms on all homotopy groups, then ''f'' is a homotopy equivalence provided ''X'' and ''Y'' are connected and have the homotopy-type of CW complexes. This result was proved by J. H. C. Whitehead in two landmark papers from 1949, and provides a justification for working with the CW complex concept that he introduced there. == Statement == More accurately, we suppose given CW complexes ''X'' and ''Y'', with respective base points ''x'' and ''y''. Given a continuous mapping : such that ''f''(''x'') = ''y'', we consider for ''n'' ≥ 1 the induced homomorphisms : where π''n'' denotes for ''n'' ≥ 1 the ''n''-th homotopy group. For ''n'' = 0 this means the mapping of the path-connected components; if we assume both ''X'' and ''Y'' are connected we can ignore this as containing no information. We say that ''f'' is a weak homotopy equivalence if the homomorphisms ''f'' * are all isomorphisms. The Whitehead theorem then states that a weak homotopy equivalence, for connected CW complexes, is a homotopy equivalence. 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「Whitehead theorem」の詳細全文を読む スポンサード リンク
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