Blakers-Massey theorem

This fact is related to: excisive triads

Statement

Let $(X;X_{1},X_{2})$ be an excisive triad, viz $X$ is a topological space and $X_{1}$, $X_{2}$ are subspaces such that the union of the interiors of $X_{i}$s is $X$). Then if $(X_{1},X_{1}\cap X_{2})$ is $(n-1)$-connected and $(X_{2},X_{1}\cap X_{2})$ is $(m-1)$-connected, the inclusion map $\pi _{q}(X_{1},X_{1}\cap X_{2})\to \pi _{q}(X,X\cap X_{2})$ is an isomorphism for $q<m+n-2$ and a surjection for $q=m+n-2$.