 Homotopy excision theorem

For a based CW triad with connected intersection $A$, if $(B,A)$ is $p$-connected and $(C,A)$ is $q$-connected, $p,q\ge 1$, the comparison map is an isomorphism for $i<p+q$ and a surjection for $i=p+q$. This connectivity form of the Blakers-Massey theorem controls the failure of ordinary relative homotopy to satisfy unrestricted excision.

The Homotopy Excision Theorem is an important result in algebraic topology that deals with the behavior of homotopy groups under certain conditions related to pairs of spaces. In essence, it allows us to conclude that if two spaces are homotopy equivalent, then certain derived spaces (like certain subspaces and their complements) retain these equivalences within homotopy categories.