MayerVietoris

Mayer-Vietoris is a fundamental tool in algebraic topology, named after Walther Mayer and Leopold Vietoris. It concerns the calculation of homology or cohomology groups of a topological space decomposed as a union of two subspaces. Given a space X = A ∪ B with A and B open (or with suitable conditions for covers), the Mayer-Vietoris sequence is a long exact sequence of abelian groups relating the homology groups of A, B, A ∩ B, and X. For homology, the sequence is … → H_n(A ∩ B) → H_n(A) ⊕ H_n(B) → H_n(X) → H_{n-1}(A ∩ B) → …; for cohomology, there is a dual sequence with connecting morphisms reversed in degree. The maps come from inclusion-induced homomorphisms and the connecting homomorphism δ arises from chain-level decompositions of chains on X with respect to the cover.

This tool is used to compute the homology of spaces built by gluing along a common subspace. It is particularly effective when A, B, and A ∩ B have simpler homology. Classic applications include proving that spheres have the expected homology, computing the homology of wedges and certain manifolds, or analyzing complex constructions like cell complexes and projective spaces obtained by gluing. The theorem extends to reduced homology, cohomology, and more general theories; it follows from excision and the long exact sequence of the pair, under mild hypotheses about the cover.