Sphere Homology via Mayer-Vietoris

Algebraic Topology · Axiom Academy

EXAMPLE Sphere Homology via Mayer-Vietoris Computing H_n(S^k) using the Mayer-Vietoris sequence with hemispheres Excellent work! You've computed the homology of spheres using Mayer-Vietoris. Here's what we learned: Strategic Covering: The Mayer-Vietoris sequence is powerful when we can cover a space with contractible open sets whose intersection is also simple. Exact Sequences: The exactness of the Mayer-Vietoris sequence allows us to deduce unknown homology groups from known ones by analyzing where maps are zero and surjective. Sphere Homology: We've proven that H_n(S^k) = Z for n = 0 or n = k, and 0 otherwise, which is fundamental in algebraic topology. Inductive Power: This method works for all dimensions - once we understand the pattern, we can compute H_*(S^k) for any k. The Mayer-Vietoris sequence is one of the most powerful computational tools in algebraic topology. Practice applying it to other spaces!

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