Corollary 5.18. (Blakers–Massey)
Assume \(f\) is \(m\)-connected and \(g\) is \(n\)-connected. Then the map \(Z \to X \times_W Y\) is \((m+n)\)-connected.
Proof
The map \(\Delta_f\) is \((m-1)\)-connected while \(\Delta_g\) is \((n-1)\)-connected. By Lemma 5.17, this means that \(\Delta_f \square_Z \Delta_g\) is \((m-1 + (n-1) + 2)\)-connected, and thus the theorem applies.