Homotopy excision theorem

In algebraic topology, the homotopy excision theorem offers a substitute for the absence of excision in homotopy theory. More precisely, let ${\displaystyle (X;A,B)}$ be an excisive triad with ${\displaystyle C=A\cap B}$ nonempty, and suppose the pair ${\displaystyle (A,C)}$ is (${\displaystyle m-1}$)-connected, ${\displaystyle m\geq 2}$, and the pair ${\displaystyle (B,C)}$ is (${\displaystyle n-1}$)-connected, ${\displaystyle n\geq 1}$. Then the map induced by the inclusion ${\displaystyle i\colon (A,C)\to (X,B)}$,

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

is bijective for ${\displaystyle q<m+n-2}$ and is surjective for ${\displaystyle q=m+n-2}$.

A geometric proof is given in a book by Tammo tom Dieck.^[1]

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. ^[2]

The most important consequence is the Freudenthal suspension theorem.