Path Lifting Property

Algebraic Topology · Axiom Academy

Unique lifting of paths from base to total space Given a covering map p: E → B, a path γ: [0,1] → B in the base space, and a starting point ẽ₀ ∈ E with p(ẽ₀) = γ(0), there exists a unique path γ̃: [0,1] → E such that: γ̃(0) = ẽ₀ (starts at the specified point) p ∘ γ̃ = γ (projects down to the original path) The proof uses the Lebesgue number lemma . Since [0,1] is compact, we can partition it into small intervals such that the image of each interval under γ lies entirely within an evenly covered neighborhood. The path γ maps the interval into some evenly covered neighborhood U We lift this piece to the unique sheet V α containing our current position The homeomorphism p| V α : V α → U determines the lift uniquely Continuity follows from gluing these local lifts together 3. Example: Lifting Paths to ℝ from S¹ Consider the covering p: ℝ → S¹ given by p(t) = e 2πit . Let γ be a path in S¹ that goes around the circle once counterclockwise, starting at 1. If we choose ẽ₀ = 0 ∈ ℝ as our starting point (since p(0) = 1), the unique lift is the straight path γ̃(t) = t in ℝ from 0 to 1. γ̃ projects to γ: p(γ̃(t)) = e 2πit = γ(t) ✓ 4. Consequences and Applications The path lifting property has profound implications: Why? Suppose [α], [β] ∈ π₁(E, ẽ₀) with p * ([α]) = p * ([β]). This means p ∘ α and p ∘ β are homotopic in B. By the homotopy lifting property (next lesson!), this homotopy lifts to a homotopy in E between α and β, so [α] = [β].

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