# 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.

## References

- T. tom Dieck,
*Algebraic Topology*, EMS Textbooks in Mathematics, (2008). - 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.

## Bibliography

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

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