Computing Higher Homotopy Groups

Algebraic Topology · Axiom Academy

EXAMPLE Computing Higher Homotopy Groups Step-by-step computations of homotopy groups using fundamental theorems Excellent work! You've completed four fundamental homotopy group computations. Here's what we learned: Hurewicz Theorem: For simply connected spaces, the first non-trivial homotopy group equals the first non-trivial homology group. This gives us immediately. Long Exact Sequence: Fibrations provide powerful computational tools. The Hopf fibration yields through exactness. Degree Theory: Self-maps of are classified by degree, establishing the fundamental isomorphism . Product Spaces: Higher homotopy groups behave well with products: , showing that the 2-sphere structure is critical for non-trivial higher homotopy. These techniques form the foundation for computing homotopy groups in algebraic topology. The interplay between fibrations, exact sequences, and fundamental theorems provides a powerful computational framework!

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