Homotopy excision theorem

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, $m\geq 2$ , and the pair $(B,C)$ is ( $n-1$ )-connected, $n\geq 1$ . Then the map induced by the inclusion $i:(A,C)\to (X,B)$

i_{*}:\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 nice geometric proof is given in the 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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[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.

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