Homotopy excision theorem

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 Tom Dieck.[1]

This result should also be seen as a consequence of the Blakers–Massey theorem, the most general form of which, dealing with the non-simply-connected case.[2]

The most important consequence is the Freudenthal suspension theorem.


  1. T. tom Dieck, Algebraic Topology, EMS Textbooks in Mathematics, (2008).
  2. R. Brown and J.-L. Loday, Homotopical excision and Hurewicz theorems for n-cubes of spaces, Proc. London Math. Soc., (3) 54 (1987) 176-192.


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