Homotopy Group - Long Exact Sequence of A Fibration

Long Exact Sequence of A Fibration

Let p: E → B be a basepoint-preserving Serre fibration with fiber F, that is, a map possessing the homotopy lifting property with respect to CW complexes. Suppose that B is path-connected. Then there is a long exact sequence of homotopy groups

... → π_n(F) → π_n(E) → π_n(B) → π_n−1(F) →... → π₀(E) → 0.

Here the maps involving π₀ are not group homomorphisms because the π₀ are not groups, but they are exact in the sense that the image equals the kernel.

Example: the Hopf fibration. Let B equal S2 and E equal S3. Let p be the Hopf fibration, which has fiber S1. From the long exact sequence

⋯ → π_n(S1) → π_n(S3) → π_n(S2) → π_n−1(S1) → ⋯

and the fact that π_n(S1) = 0 for n ≥ 2, we find that π_n(S3) = π_n(S2) for n ≥ 3. In particular, π₃(S2) = π₃(S3) = Z.

In the case of a cover space, when the fiber is discrete, we have that π_n(E) is isomorphic to π_n(B) for all n greater than 1, that π_n(E) embeds injectively into π_n(B) for all positive n, and that the subgroup of π₁(B) that corresponds to the embedding of π₁(E) has cosets in bijection with the elements of the fiber.