In algebraic topology, the homotopy excision theorem offers a substitute for the absence of excision in homotopy theory. More precisely, let be an excisive triad with nonempty, and suppose the pair is ()-connected, , and the pair is ()-connected, . Then the map induced by the inclusion ,

,

is bijective for and is surjective for .

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.

References

  1. Tammo tom Dieck, Algebraic Topology, EMS Textbooks in Mathematics, (2008).
  2. Brown, Ronald; Loday, Jean-Louis (1987). "Homotopical excision and Hurewicz theorems for n-cubes of spaces". Proceedings of the London Mathematical Society. 54 (1): 176–192. doi:10.1112/plms/s3-54.1.176. MR 0872255.

Bibliography

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