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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>
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| is bijective for <math>q < m+n-2</math> and is surjective for <math>q = m+n-2</math>.
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| A nice geometric proof is given in the book by tom Dieck.
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| 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.
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| The most important consequence is the [[Freudenthal suspension theorem]].
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| == References ==
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| * J.P. May, ''A Concise Course in Algebraic Topology'', Chicago University Press.
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| * T. tom Dieck, ''Algebraic Topology'', EMS Textbooks in Mathematics, (2008).
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| * 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.
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| [[Category:Homotopy theory]]
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| [[Category:Theorems in algebraic topology]]
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| {{topology-stub}}
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Hi here. My name is Maynard and I totally dig that status. He's always loved living in California. His wife doesn't like it the way he does but what he really likes doing is jogging and he'll be starting something else along the planet. My job a information policeman. Go to my website to find out more: http://devolro.com/expeditions
Stop by my webpage - the most