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, $m \ge 2$, and the pair $(B, C)$ is ($n-1$)-connected, $n \ge 1$. Then the map induced by 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 $q = m+n-2$.

A nice geometric proof is given in the 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.