View source for Blakers–Massey theorem

{{short description|Results on triad homotopy groups}}
In mathematics, the first '''Blakers–Massey theorem''', named after Albert Blakers and [[Biography:William S. Massey|William S. Massey]],<ref>{{Cite journal| last1=Blakers | first1=Albert L. |last2=Massey | first2=William S. | authorlink2=William S. Massey| year=1949|title=The homotopy groups of a triad|
journal=Proceedings of the National Academy of Sciences of the United States of America|volume=35|issue=6|pages=322–328|doi=10.1073/pnas.35.6.322|pmc=1063027|pmid=16588898| bibcode=1949PNAS...35..322B |mr=0030757| doi-access=free }}</ref><ref>{{Citation | last1=Blakers | first1=Albert L. |last2=Massey | first2=William S. |  title=The homotopy groups of a triad. I | journal=[[Annals of Mathematics]] |series=(2)| year=1951 | volume=53 | issue=1 | pages=161–204 | mr=0038654| doi=10.2307/1969346| jstor=1969346 }}</ref><ref>{{Citation | last=Hatcher | first=Allen | authorlink=Allen Hatcher| title= Algebraic Topology, Theorem 4.23 }}</ref> gave vanishing conditions for certain [[Excisive triad|triad]] [[Homotopy group|homotopy group]]s of [[Topological space|spaces]].

==Description of the result==
This connectivity result may be expressed more precisely, as follows. Suppose ''X'' is a [[Topological space|topological space]] which is the [[Pushout (category theory)|pushout]] of the diagram

: <math>A\xleftarrow{\ f\ } C \xrightarrow{\ g\ } B</math>,

where ''f'' is an [[N-connected space#n-connected map|''m''-connected]] map and ''g'' is ''n''-connected. Then the map of pairs

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

induces an [[Isomorphism|isomorphism]] in relative [[Homotopy group|homotopy group]]s in degrees <math>k\le (m+n-1)</math> and a surjection in the next degree.

However the third paper of Blakers and Massey in this area<ref>{{cite journal| last1=Blakers | first1=Albert L. |last2=Massey | first2=William S. |  title= The homotopy groups of a triad. III|journal= [[Annals of Mathematics]]|series=(2)|volume= 58 | issue=3 |year=1953|pages= 409–417|mr=0058971|doi=10.2307/1969744| jstor=1969744 }}</ref> determines the critical, i.e., first non-zero, triad homotopy group as a [[Tensor product of modules|tensor product]], under a number of assumptions, including some simple connectivity. This condition and some dimension conditions were relaxed in work of [[Biography:Ronald Brown (mathematician)|Ronald Brown]] and [[Biography:Jean-Louis Loday|Jean-Louis Loday]].<ref name="brown-loday">{{cite journal| authorlink1=Ronald Brown (mathematician)| last1=Brown| first1=Ronald|  last2=Loday| first2= Jean-Louis| title= Homotopical excision, and Hurewicz theorems, for ''n''-cubes of spaces| journal= [[Organization:Proceedings of the London Mathematical Society|Proceedings of the London Mathematical Society]]| series= (3)|volume= 54 |year=1987|issue= 1|pages= 176–192| mr=0872255|doi=10.1112/plms/s3-54.1.176}}</ref> 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 space|simply connected]] case, one has to use the nonabelian tensor product of Brown and Loday.<ref name="brown-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 <math>m+n</math>.

== 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 [[Biography:Charles Rezk|Charles Rezk]] in 2010.<ref>{{cite web|first=Charles|last=Rezk|title= Toposes and homotopy toposes |year=2010|url=http://www.math.uiuc.edu/~rezk/homotopy-topos-sketch.pdf|at= Prop. 8.16}}</ref>

== Fully formal proof ==

In 2013 a fairly short, fully formal proof using [[Homotopy type theory|homotopy type theory]] as a [[Philosophy:Foundations of mathematics|mathematical foundation]] and an [[Agda (programming language)|Agda variant]] as a [[Proof assistant|proof assistant]] was announced by Peter LeFanu Lumsdaine;<ref>{{cite web|url=http://ncatlab.org/nlab/show/Blakers-Massey+theorem#Lumsdaine13|title=The Blakers-Massey theorem in homotopy type theory (talk at Conference on Type Theory, Homotopy Theory and Univalent Foundations)|year=2013}}</ref> this became Theorem 8.10.2 of ''Homotopy Type Theory – Univalent Foundations of Mathematics''.<ref>{{cite book|author1=The Univalent Foundations Program|title=Homotopy type theory: Univalent foundations of mathematics|date=2013|publisher=Institute for Advanced Study|url=http://homotopytypetheory.org/book}}</ref> 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==
{{reflist}}

==External links==
* {{nlab|id=Blakers-Massey+theorem|title=Blakers–Massey theorem}}
* {{cite book|last=tom Dieck|first= Tammo|title= Algebraic Topology|series= EMS Textbooks in Mathematics|year=2008|publisher=European Mathematical Society}} Theorem 6.4.1

{{DEFAULTSORT:Blakers-Massey theorem}}
[[Category:Theorems in algebraic topology]]

{{Sourceattribution|Blakers–Massey theorem}}