# 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 ${\displaystyle i:(A,C)\to (X,B)}$

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

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

A geometric proof is given in a book by Tom Dieck.[1]

This result should also be seen as a consequence of the Blakers–Massey theorem, the most general form of which, dealing with the non-simply-connected case.[2]

The most important consequence is the Freudenthal suspension theorem.

## References

1. T. tom Dieck, Algebraic Topology, EMS Textbooks in Mathematics, (2008).
2. 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.

## Bibliography

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