Homotopy excision theorem

Short description: Offers a substitute for the absence of excision in homotopy theory

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

[math]\displaystyle{ i_*\colon \pi_q(A, C) \to \pi_q(X, B) }[/math],

is bijective for [math]\displaystyle{ q \lt m+n-2 }[/math] and is surjective for [math]\displaystyle{ q = m+n-2 }[/math].

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.

Bibliography

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