Legendre's three-square theorem: Difference between revisions

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In algebraic topology, the '''homotopy excision theorem''' offers a substitute for the absence of [[Excision theorem|excision]] in [[homotopy theory]]. More precisely, let ''X'' be a [[space (mathematics)|space]] that is union of the interiors of subspaces ''A'', ''B'' with <math>C = A \cap B</math> nonempty, and suppose a pair <math>(A, C)</math> is [[n-connected|(<math>m-1</math>)-connected]], <math>m \ge 2</math>, and a pair <math>(B, C)</math> is (<math>n-1</math>)-connected, <math>n \ge 1</math>. Then, for the inclusion <math>i: (A, C) \to (X, B)</math>,
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:<math>i_*: \pi_q(A, C) \to \pi_q(X, B)</math>
is bijective for <math>q < m+n-2</math> and is surjective for <math>q = m+n-2</math>.  
 
A nice geometric proof is given in the book by tom Dieck.  
 
This result should also be seen as a consequence of the [[Blakers-Massey_theorem]], the most general form of which, dealing with the non simply connected case, is in the paper of Brown and Loday referenced below.  
 
The most important consequence is the [[Freudenthal suspension theorem]].
 
== References ==
* J.P. May, ''A Concise Course in Algebraic Topology'', Chicago University Press.  
 
* T. tom Dieck, ''Algebraic Topology'', EMS Textbooks in Mathematics, (2008).  
 
* 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.
 
[[Category:Homotopy theory]]
[[Category:Theorems in algebraic topology]]
 
{{topology-stub}}

Latest revision as of 11:14, 7 December 2014

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