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In mathematics, the Puppe sequence is a construction of homotopy theory. Intuitively, the Puppe sequence allows us to think of homology theory as a functor that takes spaces to long-exact sequences of groups. Let ''f'':''A'' → ''B'' be a continuous map between CW complexes and let ''C''(''f'') denote a cone of f, (i.e., the cofiber of the map $f$), so that we have a (cofiber) sequence: :''A'' → ''B'' → ''C''(''f''). Now we can form Σ''A'' and Σ''B'', suspensions of A and B respectively, and also Σ''f'': Σ''A'' → Σ''B'' (this is because suspension might be seen as a functor), obtaining a sequence: : Σ''A'' → Σ''B'' → ''C''(Σ''f''). Note that suspension preserves cofiber sequences. Due to this powerful fact we know that ''C''(Σ''f'') is homotopy equivalent to Σ''C''(''f''). By collapsing ''B'' ⊆ ''C''(''f'') to a point, one has a natural map ''C''(''f'') → Σ''A''. Thus we have a sequence: : ''A'' → ''B'' → ''C''(''f'') → Σ''A'' → Σ''B'' → Σ''C''(''f''). Iterating this construction, we obtain the Puppe sequence associated to ''A'' → ''B'': : ''A'' → ''B'' → ''C''(''f'') → Σ''A'' → Σ''B'' → Σ''C''(''f'') → Σ2''A'' → Σ2''B'' → Σ2''C''(''f'') → Σ3''A'' → Σ3''B'' → Σ3''C''(''f'') → .... ==Some properties and consequences== It is a simple exercise in topology to see that every three elements of a Puppe sequence are, up to a homotopy, of the form: : ''X'' → ''Y'' → ''C''(''f''). By "up to a homotopy", we mean here that every 3 elements in a Puppe sequence are of the above form if regarded as objects and morphisms in suitable category: homotopy category. If one is now given a topological half-exact functor, the above property implies that after acting with the functor in question on the Puppe sequence associated to ''A'' → ''B'', one obtains a long exact sequence. Most notably this is the case with a family of functors of homology – the resulting long exact sequence is called the sequence of a pair (''A'',''B'') (see Eilenberg–Steenrod axioms; However, a different approach is taken in that article and a sequence of a pair is treated there as an axiom). 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「Puppe sequence」の詳細全文を読む スポンサード リンク
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