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 $$i\colon (A, C) \to (X, B)$$,
 * $$i_*\colon \pi_q(A, C) \to \pi_q(X, B)$$,

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

A geometric proof is given in a book by Tammo tom Dieck.

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

The most important consequence is the Freudenthal suspension theorem.