A long exact sequence in algebraic topology that relates the homology or cohomology of a space decomposed as the union of two subspaces A and B to the homology or cohomology of A, B, and their intersection A∩B, enabling computations via excision and algebraic gluing.

Principle

Exploit the decomposition X = A ∪ B to produce connecting homomorphisms and an exact sequence that transfers local homological data on A, B, and A∩B into global information about X.

Demonstration

For singular homology, the Mayer–Vietoris sequence yields … → H_n(A∩B) → H_n(A) ⊕ H_n(B) → H_n(X) → H_{n−1}(A∩B) → … which allows computing H_*(X) from the known groups of A, B and A∩B and the connecting maps.

Misapplication

Applying Mayer–Vietoris without ensuring the cover conditions (e.g., appropriate open or excisive pairs) or miscomputing connecting homomorphisms can produce incorrect homology groups and flawed conclusions about X.

Consequence

When applicable, the sequence reduces global homology computations to local pieces and their overlaps, producing exact algebraic constraints that determine unknown groups and support induction and van Kampen-type arguments.

Reversal

The reversal is a decomposition with no control over intersections or excision failure: without a suitable cover one cannot assemble local homological data into a coherent exact sequence for the whole space.

Boundary

Applies under hypotheses such as open covers or excisive pairs where inclusion-induced maps satisfy excision conditions; excludes arbitrary decompositions lacking the technical hypotheses required for the long exact sequence.

Semantic Tension

Tension appears with spectral-sequence approaches: both reduce global problems to local data, but Mayer–Vietoris gives a concrete long exact sequence good for low-degree computations, while spectral sequences may better handle filtrations and higher-degree complexity.

Synthesis

The Mayer–Vietoris sequence is the algebraic tool that turns a suitable cover X = A ∪ B into a long exact algebraic relation among the homology or cohomology of A, B, A∩B and X, permitting computation and structural deductions by gluing local invariants.