Editing Mayer–Vietoris sequence (section)

{{Short description|Algebraic tool for computing invariants of topological spaces}}
In [[mathematics]], particularly [[algebraic topology]] and [[homology theory]], the '''Mayer–Vietoris sequence''' is an [[algebra]]ic tool to help compute [[algebraic invariant]]s of [[topological space]]s. The result is due to two [[Austria]]n mathematicians, [[Walther Mayer]] and [[Leopold Vietoris]]. The method consists of splitting a space into [[Subspace topology|subspaces]], for which the homology or cohomology groups may be easier to compute. The sequence relates the (co)homology groups of the space to the (co)homology groups of the subspaces. It is a [[Natural (category theory)|natural]] [[long exact sequence]], whose entries are the (co)homology groups of the whole space, the [[direct sum of abelian groups|direct sum]] of the (co)homology groups of the subspaces, and the (co)homology groups of the [[intersection (set theory)|intersection]] of the subspaces.

The Mayer–Vietoris sequence holds for a variety of [[cohomology theory|cohomology]] and [[homology theory|homology theories]], including [[simplicial homology]] and [[singular cohomology]]. In general, the sequence holds for those theories satisfying the [[Eilenberg–Steenrod axioms]], and it has variations for both [[Reduced homology|reduced]] and [[Relative homology|relative]] (co)homology.  Because the (co)homology of most spaces cannot be computed directly from their definitions, one uses tools such as the Mayer–Vietoris sequence in the hope of obtaining partial information. Many spaces encountered in [[topology]] are constructed by piecing together very simple patches. Carefully choosing the two covering subspaces so that, together with their intersection, they have simpler (co)homology than that of the whole space may allow a complete deduction of the (co)homology of the space. In that respect, the Mayer–Vietoris sequence is analogous to the [[Seifert–van Kampen theorem]] for the [[fundamental group]], and a precise relation exists for homology of dimension one.