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 be a space that is union of the interiors of subspaces A, B with $C=A\cap B$ nonempty, and suppose a pair $(A,C)$ is ($m-1$)-connected, $m\geq 2$, and a pair $(B,C)$ is ($n-1$)-connected, $n\geq 1$. Then, for the inclusion $i:(A,C)\to (X,B)$,

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

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

The most important consequence is the Freudenthal suspension theorem.

References

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

• T. tom Dieck, Algebraic Topology, EMS Textbooks in Mathematics, (2008).

• 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.