Homotopy excision theorem

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 $C=A\cap B$ nonempty, and suppose the pair $(A,C)$ is ($m-1$)-connected, $m\geq 2$, and the pair $(B,C)$ is ($n-1$)-connected, $n\geq 1$. Then the map induced by the inclusion $i\colon (A,C)\to (X,B)$,

$i_{*}\colon \pi _{q}(A,C)\to \pi _{q}(X,B)$,

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

A geometric proof is given in a book by Tammo tom Dieck.

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.

The most important consequence is the Freudenthal suspension theorem.

Bibliography

• J. Peter May, A Concise Course in Algebraic Topology, Chicago University Press.