In mathematics, the first Blakers–Massey theorem, named after
Albert Blakers and
William S. Massey
William Schumacher Massey (August 23, 1920 – June 17, 2017) was an American mathematician, known for his work in algebraic topology. The Massey product is named for him. He worked also on the formulation of spectral sequences by means of exact ...
, gave vanishing conditions for certain
triad
Triad or triade may refer to:
* a group of three
Businesses and organisations
* Triad (American fraternities), certain historic groupings of seminal college fraternities in North America
* Triad (organized crime), a Chinese transnational orga ...
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, homotop ...
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 geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
which is the
pushout of the diagram
:
,
where ''f'' is an
''m''-connected map and ''g'' is ''n''-connected. Then the map of pairs
:
induces an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word i ...
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, homotop ...
s in degrees
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 and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
, under a number of assumptions, including some simple connectivity. This condition and some dimension conditions was 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 between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
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
.
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
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