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In mathematics, the first Blakers–Massey theorem, named after Albert Blakers and William S. Massey, gave vanishing conditions for certain triad homotopy groups. This connectivity result may also be expressed as that if ''X'' is the pushout of and ''f'' is ''m''-connected and ''g'' is ''n''-connected, then the map of pairs : induces an isomorphism in relative homotopy groups in degrees ''k'' ≤ (''m'' + ''n'' − 1) and a surjection in the next degree. However the third paper of Blakers and Massey in this area, referenced below, determines the critical, i.e. first non zero, triad homotopy group as a tensor product, under a number of assumptions, including some simple connectivity. This condition and some dimension conditions are relaxed in the Brown-Loday paper referenced below. Of course the algebraic result implies the connectivity result, since a tensor product is zero if one of the factors is zero. In the non simply connected case, one has to use the nonabelian tensor product introduced by Brown and Loday. The triad connectivity result can be expressed in a number of other ways, for example it says that the pushout square above behaves like a homotopy pullback up to dimension ''m'' + ''n''. == Generalization to higher toposes == The generalization of the connectivity part of the theorem from traditional homotopy theory to any other infinity-topos with an infinity-site of definition was given by Charles Rezk in 2010. 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「Blakers–Massey theorem」の詳細全文を読む スポンサード リンク
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