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In mathematics, and specifically in the field of homotopy theory, the Freudenthal suspension theorem is the fundamental result leading to the concept of stabilization of homotopy groups and ultimately to stable homotopy theory. It explains the behavior of simultaneously taking suspensions and increasing the index of the homotopy groups of the space in question. It was proved in 1937 by Hans Freudenthal. The theorem is a corollary of the homotopy excision theorem. ==Statement of the theorem== Let ''X'' be an ''n''-connected pointed space (a pointed CW-complex or pointed simplicial set). The map :''X'' → Ω(''X'' ∧ ''S''1) induces a map :π''k''(''X'') → π''k''(Ω(''X'' ∧ ''S''1)) on homotopy groups, where Ω denotes the loop functor and ∧ denotes the smash product. The suspension theorem then states that the induced map on homotopy groups is an isomorphism if ''k'' ≤ 2''n'' and an epimorphism if ''k'' = 2''n'' + 1. A basic result on loop spaces gives the relation :π''k''(Ω(''X'' ∧ ''S''1)) ≅ π''k''+1(''X'' ∧ ''S''1) so the theorem could otherwise be stated in terms of the map :π''k''(''X'') → π''k''+1(''X'' ∧ ''S''1), with the small caveat that in this case one must be careful with the indexing. 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「Freudenthal suspension theorem」の詳細全文を読む スポンサード リンク
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