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
- ... → πn(F) → πn(E) → πn(B) → πn−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
- ⋯ → πn(S1) → πn(S3) → πn(S2) → πn−1(S1) → ⋯
and the fact that πn(S1) = 0 for n ≥ 2, we find that πn(S3) = πn(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 πn(E) is isomorphic to πn(B) for all n greater than 1, that πn(E) embeds injectively into πn(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.
Read more about this topic: Homotopy Group
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