The Idea of Homotopy

Algebraic Topology · Axiom Academy

When are two maps really the same thing in topology? Let's start by drawing two different paths from point A to point B. Use the slider to explore different paths. Now watch as one path smoothly transforms into another. This smooth transformation is the key idea. At t = 0, we have the blue path. At t = 1, we have the red path. In between, we see all the intermediate paths. Different Functions, Same Shape This idea isn't just for paths - it works for any continuous functions. Choose different functions to see if they can be deformed into each other. Let's see what mathematicians mean when they say "continuous deformation" precisely. Two continuous functions and are homotopic if there exists a continuous function t ∈ [0,1] : The "time" parameter controlling the deformation H(x, 0) = f₀(x) : At time 0, we have the first function H(x, 1) = f₁(x) : At time 1, we have the second function Continuity : The transformation is smooth, with no jumps Homotopy defines when two continuous maps are "the same" topologically - not by being identical, but by being continuously deformable into each other. A homotopy H(x,t) is a smooth transformation indexed by t ∈ [0,1], giving us a continuous family of intermediate functions. Homotopy is the foundation of algebraic topology - it lets us study spaces by examining which maps can be deformed into each other.

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