##  [Mayer–Vietoris Sequence](/mayer-vietoris-sequence-2) 

 Definition

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

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

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

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

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

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

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

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

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.