The Mayer-Vietoris sequence is the homological analogue of the inclusion-exclusion principle: it computes the homology of a union X = A union B from the homology groups of A, B, and their intersection A intersect B. The long exact sequence ... -> H_n(A intersect B) -> H_n(A) + H_n(B) -> H_n(X) -> H_{n-1}(A intersect B) -> ... systematically relates these groups, and the connecting homomorphism captures how the topology of the intersection constrains the topology of the whole. It is the primary computational tool for singular homology.
The Mayer-Vietoris sequence is the most frequently used computational tool in homology. Suppose X = A union B, where A and B are open subsets (or more generally, the interiors of A and B cover X). Then there is a long exact sequence: ... -> H_n(A intersect B) -(i_*,-j_*)-> H_n(A) direct sum H_n(B) -(k_*+l_*)-> H_n(X) -partial-> H_{n-1}(A intersect B) -> ... The first map sends a class [c] in H_n(A intersect B) to the pair (i_*[c], -j_*[c]) of its images in A and B (with a sign chosen for exactness). The second map adds the images of classes in A and B to produce a class in X. The connecting homomorphism partial links the homology of X in one dimension to the homology of the intersection in one dimension lower.
The derivation of Mayer-Vietoris from excision and the long exact sequence of a pair is instructive. Start with the pair (X, A). Its long exact sequence involves H_n(X, A). Excision (with Z = X \ B, noting cl(X \ B) subset int(A) when {int(A), int(B)} covers X) gives H_n(X, A) = H_n(B, A intersect B). Now the long exact sequence of the pair (B, A intersect B) involves H_n(A intersect B) and H_n(B). Splicing these two long exact sequences together and rearranging yields the Mayer-Vietoris sequence. Understanding this derivation shows that Mayer-Vietoris is not a separate axiom but a consequence of the more fundamental excision property.
The Mayer-Vietoris sequence is most powerful when A, B, or A intersect B have simple homology. The classic example is the computation of H_*(S^n) by covering the sphere with two contractible hemispheres whose intersection is S^{n-1}. Since contractible spaces have trivial higher homology, the Mayer-Vietoris sequence collapses to isomorphisms H_k(S^n) = H_{k-1}(S^{n-1}) for k >= 2, giving the homology of all spheres by induction. For surfaces, one typically decomposes into pieces that retract to graphs or circles, and the Mayer-Vietoris sequence assembles the answer from these simple building blocks.
The Mayer-Vietoris sequence also exists for cohomology (with arrows reversed) and for reduced homology (which simplifies the low-dimensional terms). There are relative versions for pairs and versions for arbitrary coverings (the Mayer-Vietoris spectral sequence). The Euler characteristic satisfies the clean formula chi(A union B) = chi(A) + chi(B) - chi(A intersect B), which follows immediately from the exactness of the Mayer-Vietoris sequence and the additivity of the Euler characteristic on exact sequences. This "inclusion-exclusion for topology" is one of the most elegant consequences of the sequence and reinforces its role as the homological upgrade of a familiar combinatorial principle.