Mayer-Vietoris Sequence

Algebraic Topology · Axiom Academy

LESSON Mayer-Vietoris Sequence The fundamental tool for computing homology by decomposing spaces into simpler pieces The setup begins with a topological space X that can be written as the union of two open sets A and B . That is, X = A B. The key players in our story are: A B : their intersection (also open) The Mayer-Vietoris sequence relates the homology of these four spaces in a remarkable way. The sequence looks like this (for each dimension n): ... → H n (A B) → H n (A) H n (B) → H n (X) → H n-1 (A B) → ... This is an exact sequence , meaning the image of each map equals the kernel of the next. This exactness is what makes the sequence so powerful for computations. Each arrow in the Mayer-Vietoris sequence represents a homomorphism between homology groups: H n (A B) → H n (A) H n (B) : The inclusion map sends a cycle in the intersection to the "same" cycle in both A and B. H n (A) H n (B) → H n (X) : This map "glues" cycles from A and B into a cycle in X. H n (X) → H n-1 (A B) : The boundary map , which is the most mysterious. It measures the "defect" in gluing and drops dimension by 1. The boundary map is what makes this sequence "long" - it connects different dimensions and creates a powerful computational tool.

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