Homotopy excision theorem: Difference between revisions

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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 athe pair <math>(A, C)</math> is [[n-connected|(<math>m-1</math>)-connected]], <math>m \ge 2</math>, and athe pair <math>(B, C)</math> is (<math>n-1</math>)-connected, <math>n \ge 1</math>. Then, forthe map induced by the inclusion <math>i:\colon (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>.

:<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>.

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]], which deals with the non-simply-connected case. <ref>{{cite journal | last1=Brown | first1=Ronald | author1-link=Ronald Brown (mathematician)|last2=Loday | first2=Jean-Louis | author2-link=Jean-Louis Loday| title=Homotopical excision and Hurewicz theorems for ''n''-cubes of spaces | journal=[[Proceedings of the London Mathematical Society]] | volume=54 | issue=1 | year=1987 | doi=10.1112/plms/s3-54.1.176 | pages=176–192 | mr=0872255}}</ref>

The most important consequence is the [[Freudenthal suspension theorem]].

== References ==

{{reflist}}

== Bibliography ==

* [[J. Peter May]], ''A Concise Course in Algebraic Topology'', Chicago University Press.

[[Category:Theorems in homotopy theory]]

{{topology-stub}}