Homotopy Lifting Property

Algebraic Topology · Axiom Academy

LESSON Homotopy Lifting Property Lifting homotopies from base to total space with uniqueness A homotopy is a continuous family of paths, represented as a map H: [0,1] × [0,1] → B. We think of H(s,t) as the path at "time" s, evaluated at parameter t. p ∘ H̃ = H (projects down to original homotopy) This says that any homotopy downstairs can be uniquely lifted to a homotopy upstairs, given a starting point. 2. Visualizing the Square [0,1] × [0,1] We can visualize a homotopy as a square [0,1] × [0,1] mapping to B: Horizontal lines (constant s): H(s, -) represents a path from H(s,0) to H(s,1) Vertical lines (constant t): Show how the path evolves over time Left edge (t=0): All paths start at H(s,0) Right edge (t=1): All paths end at H(s,1) The homotopy lifting property says we can lift this entire square to E, preserving all structure. 3. Consequence: p₊ is Injective on π₁ The homotopy lifting property immediately implies that the induced homomorphism p₊: π₁(E, ẽ₀) → π₁(B, b₀) is injective . Proof: Suppose [α], [β] ∈ π₁(E, ẽ₀) satisfy p₊([α]) = p₊([β]). This means p ∘ α and p ∘ β are homotopic in B via some homotopy H: [0,1] × [0,1] → B. By the homotopy lifting property, H lifts to H̃: [0,1] × [0,1] → E with: H̃(0, -) is a lift of p ∘ α starting at ẽ₀, so H̃(0, -) = α (by uniqueness of path lifting) H̃(1, -) is a lift of p ∘ β starting at ẽ₀, so H̃(1, -) = β H̃ is a homotopy between α and β in E Therefore [α] = [β] in π₁(E, ẽ₀), proving injectivity.

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