Paths and Path Homotopy

Algebraic Topology · Axiom Academy

LESSON Paths and Path Homotopy Understanding paths as continuous maps and the fundamental concept of homotopy A path in a topological space X is a continuous function from the unit interval to the space: We think of the parameter t ∈ [0,1] as time, tracing out the path as t moves from 0 to 1. The animation below shows a path in a space. Two paths are homotopic if one can be continuously deformed into the other while keeping the endpoints fixed. H(s, 0) = γ₀(s) for all s ∈ [0,1] H(s, 1) = γ₁(s) for all s ∈ [0,1] H(0, t) = γ₀(0) = γ₁(0) for all t ∈ [0,1] (fixed start) H(1, t) = γ₀(1) = γ₁(1) for all t ∈ [0,1] (fixed end) The map H is called a path homotopy . We write γ₀ ≃ γ₁ when paths are homotopic. 3. Homotopy as Continuous Family A path homotopy H(s, t) can be viewed as a continuous family of paths . For each fixed t ∈ [0,1] , the function H(−, t) is a path from the initial point to the terminal point. As t varies from 0 to 1, we get a one-parameter family of paths that continuously deforms γ₀ into γ₁ . 4. Path Homotopy as Equivalence Relation Path homotopy defines an equivalence relation on the set of paths with fixed endpoints. We must verify three properties: Reflexive: Every path is homotopic to itself: γ ≃ γ via H(s,t) = γ(s) Symmetric: If γ₀ ≃ γ₁ via H , then γ₁ ≃ γ₀ via H'(s,t) = H(s,1−t) Transitive: If γ₀ ≃ γ₁ and γ₁ ≃ γ₂ , then γ₀ ≃ γ₂ by concatenating homotopies

This is the written version of the interactive lesson above. See the full Algebraic Topology course.