In algebraic topology, the homotopy excision theorem offers a substitute for the absence of excision in homotopy theory. More precisely, let ( X ; A , B ) {\displaystyle (X;A,B)} be an excisive triad with C = A ∩ B {\displaystyle C=A\cap B} nonempty, and suppose the pair ( A , C ) {\displaystyle (A,C)} is ( m − 1 {\displaystyle m-1} )-connected, m ≥ 2 {\displaystyle m\geq 2} , and the pair ( B , C ) {\displaystyle (B,C)} is ( n − 1 {\displaystyle n-1} )-connected, n ≥ 1 {\displaystyle n\geq 1} . Then the map induced by the inclusion i : ( A , C ) → ( X , B ) {\displaystyle i\colon (A,C)\to (X,B)} ,

i ∗ : π q ( A , C ) → π q ( X , B ) {\displaystyle i_{*}\colon \pi _{q}(A,C)\to \pi _{q}(X,B)} ,

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