Beyond the Fundamental Group

Algebraic Topology · Axiom Academy

INTRO Beyond the Fundamental Group Discovering why one dimension of holes isn't enough The fundamental group detects 1-dimensional loops. Click different points in this space to create loops and see what can detect. Click anywhere in the space to wrap a loop around the hole Just as uses maps from (the circle), we can study maps from higher spheres. Adjust the dimension slider to see different spheres. Let's explore the torus to see how different homotopy groups capture different information. Click the button to reveal what each detects. You might wonder: don't homology groups already detect higher-dimensional holes? Let's compare! detects loops, but spaces have holes in all dimensions. The higher homotopy groups detect n -dimensional spherical features. consists of homotopy classes of based maps , where is the n -sphere. Unlike homology, homotopy groups capture the non-abelian nature of . They also detect features that homology misses, like the fact that (the Hopf fibration!). Computing higher homotopy groups is notoriously difficult. Even for simple spaces like spheres, the answer involves deep connections to stable homotopy theory, K-theory, and beyond. This is where the adventure truly begins!

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