In algebraic topology and homology theory, the Mayer–Vietoris sequence is an algebraic tool to help compute algebraic invariants of topological spaces. The result is due to two Austrian mathematicians, Walther Mayer and Leopold Vietoris. The method consists of splitting a space into 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 long exact sequence, whose entries are the (co)homology groups of the whole space, the direct sum of the (co)homology groups of the subspaces, and the (co)homology groups of the intersection of the subspaces. The Mayer–Vietoris sequence holds for a variety of cohomology and 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 and 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.
Background, motivation, and history Similar to the fundamental group or the higher homotopy groups of a space, homology groups are important topological invariants. Although some (co)homology theories are computable using tools of linear algebra, many other important (co)homology theories, especially singular (co)homology, are not computable directly from their definition for nontrivial spaces. For singular (co)homology, the singular (co)chains and (co)cycles groups are often too big to handle directly. More subtle and indirect approaches become necessary. The Mayer–Vietoris sequence is such an approach, giving partial information about the (co)homology groups of any space by relating it to the (co)homology groups of two of its subspaces and their intersection. See § Basic versions for singular homology. The most natural and convenient way to express the relation involves the algebraic concept of exact sequences: sequences of objects (in this case groups) and morphisms (in this case group homomorphisms) between them such that the image of one morphism equals the kernel of the next. In general, this does not allow (co)homology groups of a space to be completely computed. However, because many important spaces encountered in topology are topological manifolds, simplicial complexes, or CW complexes, which are constructed by piecing together very simple patches, a theorem such as that of Mayer and Vietoris is potentially of broad and deep applicability. Walther Mayer was introduced to topology by his colleague Leopold Vietoris when attending his lectures in 1926 and 1927 at a local university in Vienna. He was told about the conjectured result and a way to its solution, and solved the question for the Betti numbers in 1929. He applied his results to the torus considered as the union of two cylinders. Vietoris later proved the full result for the homology groups in 1930, but did not express it as an exact sequence. The concept of an exact sequence only appeared in print in the 1952 book Foundations of Algebraic Topology by Samuel Eilenberg and Norman Steenrod, where the results of Mayer and Vietoris were expressed in the modern form.
Basic versions for singular homology Let X {\displaystyle X} be a topological space and A {\displaystyle A} , B {\displaystyle B} be two subspaces whose interiors cover X {\displaystyle X} . (The interiors of A {\displaystyle A} and B {\displaystyle B} need not be disjoint.) The Mayer–Vietoris sequence in singular homology for the triad ( X , A , B ) {\displaystyle (X,A,B)} is a long exact sequence relating the singular homology groups (with coefficient group the integers Z {\displaystyle \mathbb {Z} } ) of the spaces X {\displaystyle X} , A {\displaystyle A} , B {\displaystyle B} , and the intersection A ∩ B {\displaystyle A\cap B} . There is an unreduced and a reduced version.
Unreduced version For unreduced homology, the Mayer–Vietoris sequence states that the following sequence is exact:
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