Mayer–Vietoris sequence
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.
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.
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.
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.
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.
Mayer was introduced to topology by his colleague Vietoris when attending his lectures in 1926 and 1927 at a local university in Vienna.
[1] He was told about the conjectured result and a way to its solution, and solved the question for the Betti numbers in 1929.
[3][4] Vietoris later proved the full result for the homology groups in 1930 but did not express it as an exact sequence.
[5] The concept of an exact sequence only appeared in print in the 1952 book Foundations of Algebraic Topology by Samuel Eilenberg and Norman Steenrod,[6] where the results of Mayer and Vietoris were expressed in the modern form.
The Mayer–Vietoris sequence in singular homology for the triad (X, A, B) is a long exact sequence relating the singular homology groups (with coefficient group the integers Z) of the spaces X, A, B, and the intersection A∩B.
[10] An element in Hn(X) is the homology class of an n-cycle x which, by barycentric subdivision for example, can be written as the sum of two n-chains u and v whose images lie wholly in A and B, respectively.
For reduced homology there is also a Mayer–Vietoris sequence, under the assumption that A and B have non-empty intersection.
is path-connected, the reduced Mayer–Vietoris sequence yields the isomorphism where, by exactness, This is precisely the abelianized statement of the Seifert–van Kampen theorem.
[13] To completely compute the homology of the k-sphere X = Sk, let A and B be two hemispheres of X with intersection homotopy equivalent to a (k − 1)-dimensional equatorial sphere.
Since the k-dimensional hemispheres are homeomorphic to k-discs, which are contractible, the homology groups for A and B are trivial.
The Mayer–Vietoris sequence for reduced homology groups then yields Exactness immediately implies that the map ∂* is an isomorphism.
Using the reduced homology of the 0-sphere (two points) as a base case, it follows[14] where δ is the Kronecker delta.
[15] A slightly more difficult application of the Mayer–Vietoris sequence is the calculation of the homology groups of the Klein bottle X.
One uses the decomposition of X as the union of two Möbius strips A and B glued along their boundary circle (see illustration on the right).
Finally, choosing (1, 0) and (1, −1) as a basis for Z2, it follows Let X be the wedge sum of two spaces K and L, and suppose furthermore that the identified basepoint is a deformation retract of open neighborhoods U ⊆ K and V ⊆ L. Letting A = K ∪ V and B = U ∪ L it follows that A ∪ B = X and A ∩ B = U ∪ V, which is contractible by construction.
The reduced version of the sequence then yields (by exactness)[17] for all dimensions n. The illustration on the right shows X as the sum of two 2-spheres K and L. For this specific case, using the result from above for 2-spheres, one has If X is the suspension SY of a space Y, let A and B be the complements in X of the top and bottom 'vertices' of the double cone, respectively.
If Y ⊂ X and is the union of the interiors of C ⊂ A and D ⊂ B, then the exact sequence is:[19] The homology groups are natural in the sense that if
As an important special case when G is the group of real numbers R and the underlying topological space has the additional structure of a smooth manifold, the Mayer–Vietoris sequence for de Rham cohomology is where {U, V} is an open cover of X, ρ denotes the restriction map, and Δ is the difference.
The exterior derivative dωU and dωV agree on U∩V and therefore together define an n + 1 form σ on X.
For de Rham cohomology with compact supports, there exists a "flipped" version of the above sequence: where
[9] It is a fact that the singular n-simplices of X whose images are contained in either A or B generate all of the homology group Hn(X).
[26] The derivation of the Mayer–Vietoris sequence from the Eilenberg–Steenrod axioms does not require the dimension axiom,[27] so in addition to existing in ordinary cohomology theories, it holds in extraordinary cohomology theories (such as topological K-theory and cobordism).
