the Mayer-Vietoris sequence
/ MY-er vee-TOR-iss /
The Mayer-Vietoris sequence is the divide-and-conquer tool for computing cohomology. Its idea is utterly natural: if you can cover a complicated space by two simpler open pieces whose cohomology you already understand, then the cohomology of the whole is determined — up to a precise bookkeeping of how the pieces overlap. It turns a global computation into questions about two patches and their intersection.
Precisely, suppose M = U union V with U, V open. There is a long exact sequence linking the de Rham cohomologies of M, of U and V separately, and of their intersection U intersect V: ... -> H^k_dR(M) -> H^k_dR(U) (+) H^k_dR(V) -> H^k_dR(U intersect V) -> H^{k+1}_dR(M) -> ... The maps into the middle term are restrictions, the next map is the difference of restrictions, and the connecting map raising degree by one is built using a partition of unity subordinate to {U, V}. 'Exact' means at every spot the image of the incoming map equals the kernel of the outgoing map; this exactness is the rigid constraint that pins down the unknown groups from the known ones.
It is the computational workhorse of the subject: inductively, one computes the cohomology of spheres, tori, surfaces, and most manifolds by cutting them into contractible-or-simpler pieces and turning the crank. For instance, splitting S^n into two overlapping caps (each contractible, intersection homotopy-equivalent to S^{n-1}) gives the recursion that yields H^*(S^n). A practical caveat: the sequence only constrains the groups; extracting an actual answer requires understanding the maps (especially the connecting homomorphism), and exactness alone may leave a genuine extension ambiguity in more refined (e.g. integral) theories — though over the reals, vector spaces split, which makes the de Rham version especially clean.
Compute H^*(S^1) by Mayer-Vietoris. Cover the circle by two open arcs U, V, each contractible (so each has cohomology R in degree 0, zero above). Their intersection U intersect V is two disjoint arcs, with H^0 = R^2. The sequence 0 -> H^0(S^1) -> R (+) R -> R^2 -> H^1(S^1) -> 0 forces, by exactness and dimension counting, H^0(S^1) = R and H^1(S^1) = R. The single 'extra' dimension in the intersection becomes the loop class in H^1.
Cutting S^1 into two arcs and running the exact sequence recovers H^0 = H^1 = R; the same induction gives every sphere.
Mayer-Vietoris is an exact sequence, not a formula: it relates groups but you must chase the maps (especially the degree-raising connecting map, which encodes the gluing) to get numbers. The same machine exists for singular homology and many other theories — exactness, not the specific objects, is what makes it work.