Homotopy excision theorem explained

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.^[1]

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. ^[2]

The most important consequence is the Freudenthal suspension theorem.

Bibliography

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

Notes and References

1. [Tammo tom Dieck]
2. Brown . Ronald . Ronald Brown (mathematician). Loday . Jean-Louis . Jean-Louis Loday. Homotopical excision and Hurewicz theorems for n-cubes of spaces . . 54 . 1 . 1987 . 10.1112/plms/s3-54.1.176 . 176–192 . 0872255.