Relative homotopy of a Serre fibration (${\pi }_n(E,F)\cong {\pi }_n(B)$)

For a based Serre fibration with fibre $F$ over the basepoint, lift a based cube in $B$ from its constant bottom face, keeping the side faces fixed. Its final face lies in $F$, giving surjectivity of the relative map. Homotopies in $B$ lift; two lifts of the same cube are relatively homotopic by lifting with both side liftings prescribed. This gives injectivity. For $n\ge 2$ this is a group isomorphism; at $n=1$ it is initially a bijection of pointed sets, unless the relative set is endowed with the group law transported through the fibration.