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