Blakers–Massey theorem

In mathematics, the first Blakers–Massey theorem, named after Albert Blakers and William S. Massey,^[1][2] gave vanishing conditions for certain triad homotopy groups. This connectivity result may also be expressed as that if X is the pushout of $A\stackrel{f}{\leftarrow} C\stackrel{g}{\rightarrow} B$ and f is m-connected and g is n-connected, then the map of pairs

$(A,C)\rightarrow (X,B) \,$

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.

Fully formal proof

In 2013 a fairly short fully formal proof using homotopy type theory as a mathematical foundation and an Agda variant as a proof assistant was announced by Peter LeFanu Lumsdaine . It became theorem 8.10.2 of Homotopy Type Theory – Univalent Foundations of Mathematics.^[3] This induces an internal proof for any infinity-topos (i.e. without reference to a site of definition). In particular it gives a new proof of the original result.

References

• Blakers–Massey theorem in nLab

• 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

1. Blakers, A. L.; Massey, W. S. (1951), "The homotopy groups of a triad I", Annals of Mathematics 53: 161–204, doi:10.2307/1969346
2. Hatcher, A., Algebraic Topology, Theorem 4.23
3. The Univalent Foundations Program (2013). Homotopy type theory: Univalent foundations of mathematics. Institute for Advanced Study.

• Charles Rezk, prop. 8.16 in Toposes and homotopy toposes (2010) (pdf)