Homotopy excision 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 <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>Ii\colon (A, C) \to (X, B)</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>.