homotopy excision theorem

{{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 in homotopy theory. More precisely, let $(X;\; A,\; B)$ be an excisive triad with $C\; =\; A\; \backslash cap\; B$ nonempty, and suppose the pair $(A,\; C)$ is ($m-1$)-connected, $m\; \backslash ge\; 2$, and the pair $(B,\; C)$ is ($n-1$)-connected, $n\; \backslash ge\; 1$. Then the map induced by the inclusion $i\backslash colon\; (A,\; C)\; \backslash to\; (X,\; B)$,

:$i\_*\backslash colon\; \backslash pi\_q(A,\; C)\; \backslash to\; \backslash pi\_q(X,\; B)$,

is bijective for and is surjective for $q\; =\; m+n-2$.

A geometric proof is given in a book by Tammo tom Dieck.Tammo tom Dieck, Algebraic Topology, EMS Textbooks in Mathematics, (2008).

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. {{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}}

The most important consequence is the Freudenthal suspension theorem.