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In mathematics, the first '''Blakers–Massey theorem''', named after [[Albert Blakers]] and [[William S. Massey]],<ref>{{Citation | last1=Blakers | first1=A. L. |last2=Massey | first2=W. S. | title=The homotopy groups of a triad I | journal=Annals of Mathematics | year=1951 | volume=53 | pages=161–204}}</ref><ref>{{Citation | last1=Hatcher | first1=A.  | title= Algebraic Topology, Theorem 4.23 }}</ref> gave vanishing conditions for certain triad homotopy groups. This connectivity result may also be expressed as that if ''X'' is the [[pushout]] of <math>A\stackrel{f}{\leftarrow} C\stackrel{g}{\rightarrow} B</math> and ''f'' is ''m''-connected and ''g'' is ''n''-connected, then the map of pairs

: <math>(A,C)\rightarrow (X,B) \, </math>

induces an [[isomorphism]] in relative [[homotopy group]]s in degrees ''k''&nbsp;≤&nbsp;(''m''&nbsp;+&nbsp;''n''&nbsp;−&nbsp;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&nbsp;''m''&nbsp;+&nbsp;''n''. 

==References==

* {{nlab|id=Blakers-Massey+theorem|title=Blakers-Massey theorem}} 

* Blakers, A. L. and Massey, W. S., The homotopy groups of a triad. {III}, Ann. of Math. (2), 58: (1953) 409–417.

* R. Brown and J.-L. Loday, Homotopical excision, and Hurewicz theorems, for n-cubes of spaces, Proc. London Math. Soc. (3) 54 (1987) 176-192.

* tom Dieck, T., Algebraic Topology, EMS Textbooks in Mathematics, (2008). Theorem 6.4.1 
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[[Category:Theorems in algebraic topology]]