Homotopy of Paths

Topology · Axiom Academy

Understanding continuous deformations in algebraic topology Imagine you have two paths in space that start at the same point and end at the same point. Can you continuously deform one path into the other without breaking it or moving the endpoints? If so, these paths are homotopic . Step 2: Formal Definition of Path Homotopy Let be two continuous paths in a topological space with the same initial point and terminal point . A path homotopy from to is a continuous map We write if such a homotopy exists. Step 3: Properties of Path Homotopy Path homotopy is an equivalence relation , which means it satisfies three important properties: Reflexive: (every path is homotopic to itself) Step 4: Homotopy Relative to a Subset Two paths are homotopic relative to if there exists a homotopy such that for each fixed , the path has the same values as and on the set . For path homotopy, we typically take , meaning the endpoints remain fixed throughout the deformation. Two topological spaces are homotopy equivalent (written ) if there exist continuous maps The maps and are called homotopy equivalences , and each is a homotopy inverse of the other. Contractible spaces: A space is contractible if it is homotopy equivalent to a point. Examples: , any convex subset of Euclidean space. Circle ≃ Annulus: A circle is homotopy equivalent to an annulus (ring). Both can be continuously deformed to each other by "thickening" or "shrinking."

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