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 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:\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>.

A nice geometric proof is given in thea 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 [[Blakers-Massey_theorem]], the most general form of whichthe [[Blakers–Massey theorem]], dealingwhich deals with the non -simply -connected case,. is<ref>{{cite injournal the| paperlast1=Brown of| first1=Ronald | author1-link=Ronald Brown and(mathematician)|last2=Loday | first2=Jean-Louis | author2-link=Jean-Louis Loday| referencedtitle=Homotopical belowexcision 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}}

* J.P. May, ''A Concise Course in Algebraic Topology'', Chicago University Press.

== Bibliography ==

* T. tom Dieck, ''Algebraic Topology'', EMS Textbooks in Mathematics, (2008).

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

[[Category:Theorems in algebraichomotopy topologytheory]]

* 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.

[[Category:Homotopy theory]]

[[Category:Theorems in algebraic topology]]

{{topology-stub}}