The Mayer-Vietoris Theorem gives a long exact sequence relating the homology groups of a topological space to the homology groups of two open subsets that cover it and of their intersection, making it possible to compute the homology of a complicated space by decomposing it into simpler pieces. Named for Walther Mayer and Leopold Vietoris, it is a basic computational tool of algebraic topology.
Facts
Partially Attested
Proof YearMayer obtained the Betti-number version in 1929; Vietoris proved the full homology-group result in 1930; the exact-sequence formulation used today came from Eilenberg and Steenrod's 1952 book, per the Background, motivation, and history section. StatementThe Mayer-Vietoris theorem gives a long exact sequence relating the homology or cohomology groups of a space to those of two open subsets whose union is the space and whose intersection is known, letting the invariants of a complicated space be computed from simpler pieces. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Mayer-Vietoris Theorem (Wikipedia)
Sources
1. Mayer-Vietoris Theorem (Wikipedia)
Wikimedia Foundationlead paragraph, first sentence
In algebraic topology and homology theory, the Mayer-Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces.
Background, motivation, and history section, the Mayer/Vietoris/Eilenberg-Steenrod paragraph
Vietoris later proved the full result for the homology groups in 1930, but did not express it as an exact sequence.
- In Branch: Algebraic Topology, Lead sentence
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