the long exact sequence of a fibration
A fibration ties three spaces together: a fiber F sitting inside a total space E that projects to a base B. The long exact sequence is the precise accounting of how their homotopy groups interlock — an infinite chain of groups and maps in which the homotopy of the fiber, the total space, and the base flow into one another with no leftover. It is the single most-used engine for actually computing homotopy groups.
Given a (Serre) fibration F -> E -> B with chosen basepoints, there is a long exact sequence ... -> pi_n(F) -> pi_n(E) -> pi_n(B) -> pi_{n-1}(F) -> pi_{n-1}(E) -> ... ending ... -> pi_1(B) -> pi_0(F) -> pi_0(E) -> pi_0(B). The maps pi_n(F) -> pi_n(E) and pi_n(E) -> pi_n(B) are induced by inclusion of the fiber and the projection; the connecting map pi_n(B) -> pi_{n-1}(F) is the boundary, built by lifting an n-sphere's worth of homotopy in the base and reading the obstruction in the fiber. 'Exact' means at every spot the image of the incoming map equals the kernel of the outgoing one — so the sequence has no gaps, and knowing any two consecutive groups plus the maps pins down a lot about the third. The construction is itself an application of the homotopy lifting property defining a fibration.
Using it is a craft: feed in the homotopy groups you know, exploit exactness to force the unknown ones. If E is contractible (all pi_n(E) = 0), the sequence collapses to isomorphisms pi_n(B) ≅ pi_{n-1}(F) — this is exactly how the path-loop fibration gives pi_n(X) = pi_{n-1}(Omega X) and how the Hopf fibration yields pi_3(S^2) = Z. Two honest caveats: the tail near pi_0 and pi_1 is only an exact sequence of pointed sets and groups, not of abelian groups, so 'exactness' there is weaker (kernels and images of maps that need not be homomorphisms), and the whole sequence depends on a basepoint and, in the non-simply-connected case, on the pi_1-action on the fiber's homotopy — it is not a free lunch that ignores basepoints.
From the fibration SO(n) -> SO(n+1) -> S^n (rotation groups acting on the sphere), the long exact sequence lets you compute pi_1 of rotation groups inductively: it yields pi_1(SO(n)) = Z/2 for n >= 3, the algebraic source of the existence of spin groups and the 'belt trick.'
The rotation-group fibration plus exactness yields pi_1(SO(n)) = Z/2 — why spin groups exist.
Exactness at the low end (pi_1, pi_0) is only of pointed sets, not groups, so you cannot blindly 'cancel' there as you would with abelian groups; and the connecting map's correctness relies on the fibration's HLP, which is why the result is false for a general surjection.