Legendre's three-square theorem

In algebraic topology, the homotopy excision theorem offers a substitute for the absence of excision in homotopy theory. More precisely, let X be a space that is union of the interiors of subspaces A, B with $C=A\cap B$ nonempty, and suppose a pair $(A,C)$ is ($m-1$)-connected, $m\geq 2$, and a pair $(B,C)$ is ($n-1$)-connected, $n\geq 1$. Then, for 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.

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, is in the paper of Brown and Loday referenced below.

The most important consequence is the Freudenthal suspension theorem.

References

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

• 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.