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 ∩ B nonempty, and suppose the pair ( A , C ) is ( m − 1 )-connected, m ≥ 2 , and the pair ( B , C ) is ( n − 1 )-connected, n ≥ 1 . Then the map induced by the inclusion i : ( A , C ) → ( 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.

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.

The most important consequence is the Freudenthal suspension theorem.