Deforming Paths

Algebraic Topology · Axiom Academy

Watching paths continuously transform into each other Here are two different paths from point A to point B in the plane. Move the slider to watch one path transform into the other. Watch how the path smoothly transforms without breaking or jumping. The endpoints stay fixed throughout! Let's see the deformation from a different perspective. As t varies, the paths "sweep out" a region. What happens if we reverse the direction of a path? Explore how path reversal affects homotopy. Red dot: travels along path from A to B Blue dot: travels the reverse path from B to A When paths connect end-to-end, we can compose them into a single path. First path (red): A → B | Second path (blue): B → C | Composed path (purple): A → C Path homotopy requires that endpoints remain fixed throughout the deformation. If γ₀, γ₁ go from A to B, then H(0,t) = A and H(1,t) = B for all t ∈ [0,1]. The homotopy H(s,t) gives a continuous family of paths. For each fixed t, we get a path from A to B. As t varies from 0 to 1, these paths transform from γ₀ to γ₁. Path homotopy classes can be composed and reversed, giving them an algebraic structure. This is the foundation for defining the fundamental group!

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