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A fibration p : E -> B with fiber F gives rise to a long exact sequence of homotopy groups: ... -> pi_n(F) -> pi_n(E) -> pi_n(B) -> pi_{n-1}(F) -> ... -> pi_1(B) -> pi_0(F) -> pi_0(E). This sequence relates the homotopy groups of the total space, base, and fiber, and is the primary computational tool for homotopy groups. It generalizes the long exact sequence of a covering space (where the fiber is discrete) and is analogous to the long exact sequence of a pair in homology.
A fibration is a continuous map p : E -> B satisfying the homotopy lifting property (HLP): given any homotopy h : X x [0,1] -> B and a lift h_0 : X -> E of the starting map (p compose h_0 = h(-, 0)), there exists a homotopy H : X x [0,1] -> E lifting the entire homotopy (p compose H = h). Intuitively, any continuous deformation in the base can be "tracked" in the total space. The fiber F = p-1(b_0) over a basepoint b_0 is the preimage, and under mild conditions (B path-connected), all fibers are homotopy equivalent.
The long exact sequence of a fibration is the analog of the long exact sequence of a pair for homotopy groups. For a fibration F -> E -> B with connected base B, there is an exact sequence: ... -> pi_n(F) -i_*-> pi_n(E) -p_*-> pi_n(B) -partial-> pi_{n-1}(F) -> ... -> pi_1(E) -p_*-> pi_1(B) -partial-> pi_0(F) -> pi_0(E). The maps i_* and p_* are induced by the inclusion F hookrightarrow E and the projection p : E -> B. The connecting homomorphism partial : pi_n(B) -> pi_{n-1}(F) is constructed using the homotopy lifting property: given a based map f : Sn -> B representing a class in pi_n(B), lift the induced map Dn -> B to a map Dn -> E. The restriction of this lift to the boundary Sn-1 = boundary(Dn) lands in the fiber F, giving a class in pi_{n-1}(F).
The most celebrated application is to the Hopf fibration S1 -> S3 -> S2. The long exact sequence reads: ... -> pi_n(S1) -> pi_n(S3) -> pi_n(S2) -> pi_{n-1}(S1) -> ... Since pi_k(S1) = 0 for k >= 2 (the universal cover R is contractible), the sequence gives isomorphisms pi_n(S3) = pi_n(S2) for n >= 3. In particular, pi_3(S2) = pi_3(S3) = Z. This is how the nontrivial pi_3(S2) is computed: not by direct construction, but by recognizing S2 as the base of the Hopf fibration and using the exact sequence to transfer the computation to the simpler space S3.
Covering spaces are a special case: a covering p : E -> B is a fibration with discrete fiber F. Since discrete spaces have trivial higher homotopy groups (pi_n(F) = 0 for n >= 1), the long exact sequence gives isomorphisms pi_n(E) = pi_n(B) for n >= 2 and a short exact sequence 1 -> pi_1(E) -> pi_1(B) -> pi_0(F) for n = 1. This recovers the classical theory: a covering space has the same higher homotopy groups as the base, and its fundamental group is a subgroup of pi_1(B) with index equal to the number of sheets.
The path-loop fibration is another fundamental example: for any pointed space (B, b_0), the path space PB = {paths in B starting at b_0} is contractible, and the endpoint evaluation map PB -> B is a fibration with fiber Omega B = {loops in B based at b_0} (the loop space). Since PB is contractible, the long exact sequence gives isomorphisms pi_n(B) = pi_{n-1}(Omega B) for all n >= 1. This "looping" operation shifts homotopy groups down by one dimension and is one of the most important structural results in homotopy theory. It underpins the theory of iterated loop spaces, spectra, and stable homotopy theory — the modern framework for understanding the deep structure of homotopy groups.