Fibration connecting map

Let $p:(E,e_0)\to (B,b_0)$ be a based Serre fibration and let $F=p^{-1}(b_0)$, with basepoint $e_0$, as in Fiber and fiber homotopy equivalence. For $n\ge 1$, represent $[b]\in {\pi }_n(B,b_0)$ by a based cube $b:I^n\to B$ as in Higher homotopy group by based cubes. Write $u\in I^{n-1}$ and $t\in I$ for its coordinates. The distinguished face is $t=0$; the union $J$ of all other faces is the relative convention of Relative homotopy classes and groups.

Apply A fibration has path lifting and homotopy lifting relative to a subspace to the homotopy $b(u,1-s)$, starting with the constant $e_0$ lift at $s=0$ and keeping $u\in \mathrm{\partial }{I}^{n-1}$ fixed. Reversing $s$ gives a lift $\tilde{b}:I^n\to E$ with $p\tilde{b}=b$ and $\tilde{b}{\mathrm{\mid }}_J=e_0$. Its face $a(u)=\tilde{b}(u,0)$ lies in $F$ and is based on $\mathrm{\partial }{I}^{n-1}$.

The connecting map is ${\mathrm{\partial }}_p[b]=[a]\in {\pi }_{n-1}(F,e_0)(n\ge 2)\mathrm{.}$ For $n=1$ it is the path component $[\tilde{b}(0)]\in {\pi }_0(F,e_0)$. Thus a loop is lifted with its terminal point fixed to $e_0$, and the initial component is recorded. This orientation matters: it need not equal the endpoint component of a lift with initial point $e_0$.

Independence of representative and lift and the homomorphism property for $n\ge 2$ are proved in the next lemma, the declared justifier. No group structure on ${\pi }_0(F)$ is assumed. Only finite cubical relative lifting is used here, so AC is unnecessary. The fiber is nonempty because $e_0$ is specified.