The Fundamental Group (very basic)

GRE Math Subject · Axiom Academy

LESSON The Fundamental Group (Very Basic) Homotopy of paths, the group structure of loops, and the few results the GRE actually tests Definition: Let be a topological space. A path from to is a continuous map with and . Definition: Two paths from to are path-homotopic (written ) if there exists a continuous map such that: The map is called a path homotopy . The endpoints stay fixed throughout the deformation. Intuition: A path homotopy continuously deforms one path into another while keeping the endpoints pinned. Think of it as stretching a rubber band from one shape to another, with the two ends nailed down. Definition: A loop based at is a path with . The fundamental group of at the basepoint , denoted , is the set of path-homotopy classes of loops based at , equipped with the operation of concatenation . Concatenation: Given loops based at , define by: In other words: traverse at double speed, then traverse at double speed. Group axioms (all verified up to homotopy): Identity: The constant loop satisfies Inverses: The reverse path satisfies Step 3: Simply Connected Spaces Definition: A space is simply connected if it is path-connected and . Equivalently: is simply connected if it is path-connected and every loop can be continuously shrunk to a point. for any — every loop can be contracted (no "holes") for — the sphere has no holes a loop can wrap around The open disk, any star-shaped region — a loop around the origin cannot be shrunk to a point

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