Long exact homotopy sequences

Let $p:E\to B$ be a Serre fibration, or a Hurewicz fibration, choose $e_0\in E$, and put $b_0=p(e_0)$ and $F=p^{-1}(b_0)$. Assume B path connected for the displayed tail; otherwise restrict to the path component of $b_0$ and its inverse image. The sequence has maps induced by inclusion and projection, together with connecting maps $\partial:\pi_n(B,b_0)\to\pi_{n-1}(F,e_0)$. For n at least two these are group homomorphisms. The based groups of F and E concern the components of their chosen points even if the entire spaces are disconnected.

To picture the connecting map, represent a based sphere by a cube with boundary sent to $b_0$. Lift from prescribed faces using homotopy lifting. The remaining boundary face lands in F and represents the lower-dimensional class. Choices change the representative by a based homotopy, so the resulting class is well-defined. For n equal to one, the lifted loop ends somewhere in F, and the connecting map records its path component. This endpoint-component description is a map to a pointed set, not in general a homomorphism into a group.

At a group term, exactness says image equals kernel. For example, a base loop lifts to a loop up to a path inside F precisely when its class lies in the image of $\pi_1(E,e_0)$. At a pointed-set term, replace kernel by the preimage of the distinguished element. Thus the components of F mapping into the component of $e_0$ in E are exactly the endpoint components reached by lifted loops from $e_0$. There is no reason for the component set of an arbitrary fiber to possess an addition operation.

Monodromy gives a fuller description: $\pi_1(B,b_0)$ acts on the path components of F, and its orbits correspond to path components of E when B is path connected. If E is path connected, that action is transitive, and the stabilizer of the distinguished fiber component is the image of $\pi_1(E,e_0)$. This statement generalizes the coset description for covering spaces. It does not require F itself to be path connected. Ignoring the action can lose information about components even when every higher group shown in an attempted calculation vanishes.

If both neighbors of a connecting homomorphism vanish in the appropriate positions, exactness can force it to be an isomorphism. More generally, a segment $0\to A\to B\to C\to0$ records a group extension and does not automatically identify B with a direct product. The maps carry information beyond the names of the groups. The path-space fibration, whose total space is contractible, yields $\pi_n(B)\cong\pi_{n-1}(\Omega B)$ for n at least two and identifies components of the loop space with based loop classes at the tail.

The Hopf bundle provides a geometric example with circle fiber, three-sphere total space, and two-sphere base. Its projection sends a unit complex pair $(z_0,z_1)$ to the complex line it spans, identifying the base with $\mathbb CP^1\cong S^2$. Fibers are common complex-phase circles, and standard local charts show it is a bundle, hence a fibration in this setting. The exact sequence explains nontrivial higher sphere maps without claiming the total space is a product. Choosing generators and orientations fixes signs in connecting maps; an unspecified orientation gives an isomorphism only up to sign.