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{{short description|Offers a substitute for the absence of excision in homotopy theory}}
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> :<math>i_*\colon \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>.
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A geometric proof is given in a book by [[Tammo tom Dieck]].<ref>[[Tammo tom Dieck]], ''Algebraic Topology'', EMS Textbooks in Mathematics, (2008).</ref>
This result should also be seen as a consequence of the most general form of the [[Blakers–Massey theorem]],
The most important consequence is the [[Freudenthal suspension theorem]].
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* [[J. Peter May]], ''A Concise Course in Algebraic Topology'', Chicago University Press.
[[Category:
{{topology-stub}}
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