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In mathematics, the first Blakers–Massey theorem, named after
Albert Blakers Albert may refer to: Companies * Albert Computers, Inc., a computer manufacturer in the 1980s * Albert Czech Republic, a supermarket chain in the Czech Republic * Albert Heijn, a supermarket chain in the Netherlands * Albert Market, a street mark ...
and William S. Massey, gave vanishing conditions for certain triad
homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
s of spaces.


Description of the result

This connectivity result may be expressed more precisely, as follows. Suppose ''X'' is a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
which is the pushout of the diagram : A\xleftarrow C \xrightarrow B, where ''f'' is an ''m''-connected map and ''g'' is ''n''-connected. Then the map of pairs : (A,C)\rightarrow (X,B) induces an
isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
in relative
homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
s in degrees k\le (m+n-1) and a surjection in the next degree. However the third paper of Blakers and Massey in this area determines the critical, i.e., first non-zero, triad homotopy group as a
tensor product In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of ...
, under a number of assumptions, including some simple connectivity. This condition and some dimension conditions were relaxed in work of Ronald Brown and Jean-Louis Loday. 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 In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every Path (topology), path between two points can be continuously transformed into any other such path while preserving ...
case, one has to use the nonabelian tensor product of 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 In mathematical logic and computer science, homotopy type theory (HoTT) refers to various lines of development of intuitionistic type theory, based on the interpretation of types as objects to which the intuition of (abstract) homotopy theory ap ...
as a mathematical foundation and an Agda variant as a
proof assistant In computer science and mathematical logic, a proof assistant or interactive theorem prover is a software tool to assist with the development of formal proofs by human–machine collaboration. This involves some sort of interactive proof edi ...
was announced by Peter LeFanu Lumsdaine; this became Theorem 8.10.2 of ''Homotopy Type Theory – Univalent Foundations of Mathematics''. 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


External links

* * Theorem 6.4.1 {{DEFAULTSORT:Blakers-Massey theorem Theorems in algebraic topology