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 a pair ${\displaystyle (A,C)}$ is (${\displaystyle m-1}$)-connected, ${\displaystyle m\geq 2}$, and a pair ${\displaystyle (B,C)}$ is (${\displaystyle n-1}$)-connected, ${\displaystyle n\geq 1}$. Then, for the inclusion ${\displaystyle i:(A,C)\to (X,B)}$,

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

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

The most important consequence is the Freudenthal suspension theorem.

References

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