fibration

A fibration is a continuous map p: E → B that satisfies a lifting property with respect to homotopies. A Hurewicz fibration has the homotopy lifting property for every space X: given a map f: X × I → B and a lift f0: X → E of f restricted to X × {0}, there exists a lift F: X × I → E with F|_{X×{0}} = f0 and p ∘ F = f. A Serre fibration requires this lifting property only for all CW complexes X. In general, every Hurewicz fibration is a Serre fibration, but the converse need not hold.

For a fibration p: E → B and a chosen base point b ∈ B, the fiber over b is F_b = p^{-1}(b). The existence of the lifting property yields a fundamental homotopical tool: the long exact sequence of homotopy groups, linking the homotopy groups of F_b, E, and B via ... → π_n(F_b) → π_n(E) → π_n(B) → π_{n-1}(F_b) → ..., valid for n ≥ 2, with an action of π1(B) on π_n(F_b).

Examples and relations: the projection p: E × F → E is a trivial fibration with fiber {e} × F ≅ F. The Hopf fibration S^3 → S^2 has fiber S^1 and is a nontrivial example of a fibration. Covering maps are fibrations with discrete fibers. A fiber bundle is a fibration equipped with local triviality: every point b ∈ B has a neighborhood U with p^{-1}(U) ≅ U × F.

Fibrations are central in algebraic topology because they organize how the homotopy types of E, B, and F interrelate, enabling computation of homotopy and cohomology information through the long exact sequence and related tools.