Long Exact Sequence of a Fibration

Algebraic Topology · Axiom Academy

LESSON Long Exact Sequence of a Fibration A powerful tool for computing homotopy groups using fibration sequences 1. Fibrations and the Homotopy Lifting Property A continuous map p : E → B is a fibration if it has the homotopy lifting property (HLP): for any space X , given a homotopy H : X × [0,1] → B and a map h : X → E such that p ∘ h = H (−,0), there exists a lift H̃ : X × [0,1] → E with p ∘ H̃ = H and H̃ (−,0) = h . The fiber over a point b ∈ B is F = p −1 ( b ). For path-connected base spaces, all fibers are homotopy equivalent. Covering spaces: Every covering map is a fibration Vector bundles: The projection of any vector bundle is a fibration Hopf fibration: S³ → S² with fiber S¹ For a fibration p : E → B with fiber F over a basepoint b ₀, we obtain an infinite exact sequence of homotopy groups: i * : induced by the inclusion i : F ↪ E p * : induced by the projection p : E → B ∂ : the connecting homomorphism (boundary map) 3. The Connecting Homomorphism The connecting homomorphism ∂ : π n ( B , b ₀) → π n −1 ( F , e ₀) is the most subtle part of the sequence. It is constructed using the homotopy lifting property: Start with a based map α : ( S n , *) → ( B , b ₀) representing [ α ] ∈ π n ( B ) View S n as D n / ∂D n ; lift the constant map on ∂D n to e ₀ ∈ E Use HLP to extend this to a lift α̃ : ∂D n → E The restriction α̃ | ∂D n gives a map S n −1 → F Define ∂[ α ] = [ α̃ | ∂D n ] ∈ π n −1 ( F )

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