Homotopy and π₁ Summary

Algebraic Topology · Axiom Academy

Let's review path homotopy, the fundamental group, and how π₁ detects holes in spaces. Definition: Two paths with the same endpoints are homotopic if one can be continuously deformed into the other while keeping endpoints fixed Formal Condition: A homotopy is a continuous map where Equivalence Relation: Path homotopy is reflexive, symmetric, and transitive, partitioning paths into equivalence classes Geometric Intuition: Paths that can be "slid around" obstacles in the same way are homotopic The Fundamental Group π₁(X, x₀) Elements: Homotopy classes of loops based at Group Operation: Path concatenation defined by traversing then Inverses: The reverse path traverses the same path backwards First Algebraic Invariant: Homotopy equivalent spaces have isomorphic fundamental groups Example Recap: Computing π₁(S¹) Step 1 - Identify Loops: Each loop in the circle winds around some integer number of times (winding number) Step 2 - Homotopy Classes: Two loops are homotopic if and only if they have the same winding number Step 3 - Group Structure: Concatenating loops adds winding numbers: a loop winding m times followed by one winding n times gives total winding m + n Step 4 - Conclusion: — the circle's fundamental group is the integers under addition Simply Connected Spaces: for contractible spaces like , disks, and convex sets Circle: detects the "hole" in the circle Figure-Eight: , the free group on two generators Torus: , capturing two independent loops

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