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#REDIRECT [[Homotopy excision theorem]]
In [[algebraic topology]], the '''homotopy excision theorem''' offers a substitute for the absence of [[Excision theorem|excision]] in [[homotopy theory]]. More precisely, let <math>(X; A, B)</math> be an [[excisive triad]] with <math>C = A \cap B</math> nonempty, and suppose the pair <math>(A, C)</math> is [[n-connected|(<math>m-1</math>)-connected]], <math>m \ge 2</math>, and the pair <math>(B, C)</math> is (<math>n-1</math>)-connected, <math>n \ge 1</math>. Then the map induced by the inclusion <math>i: (A, C) \to (X, B)</math>
:<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.<ref>T. tom Dieck, ''Algebraic Topology'', EMS Textbooks in Mathematics, (2008).</ref>
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.<ref>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.</ref>
The most important consequence is the [[Freudenthal suspension theorem]].
== References ==
{{reflist}}
== Bibliography ==
* J.P. May, ''A Concise Course in Algebraic Topology'', Chicago University Press.
[[Category:Homotopy theory]]
[[Category:Theorems in algebraic topology]]
{{topology-stub}}
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