Mayer-Vietoris sequence

For a topological space X, the homology groups H_n(X) are generally much easier to compute than the homotopy groups π_n(X), and consequently we usually will have an easier time working with such a group to aid in the classification of spaces. Homology groups are Abelian, and so computations can often be done directly using the tools of linear algebra (in simplicial homology). However, even so, eventually such computations become cumbersome, and it is useful to have tools that allow us to compute homology groups from others we already know (this approach, of course, is used everywhere in mathematics). One of the most useful tools for this is the Mayer-Vietoris sequence (it is somewhat analogous to the Seifert-Van Kampen theorem in homotopy theory).

For X a topological space with two open subsets U and V whose union is X, the Mayer-Vietoris sequence is a long exact sequence which relates the (singular) homology groups of the space X to those of U, V, and their intersection. The sequence is as follows:

where the maps between each homology group of the same n are various combinations of homomorphisms induced by of U ∩ V into U and V, and U and V into X. The mapping into the direct sum is just the product map, and the map into H_n(X) is the difference of the two homomorphisms. The maps that "step down" from n+1 to n and n to n-1 are the boundary maps ∂.

One of the most immediate applications of this is proving that the kth homology group of the sphere Sⁿ is trivial unless k = 0 or k = n, in which case H_k(Sⁿ) is isomorphic to Z. Such a complete classification of the homology groups for spheres starkly contrasts with what is known for homotopy groups of spheres π_k(Sⁿ) (for k ≤ n things are good, but not much is known in the case k > n). See homology theory for more.

Still working on this. Or, if you like, you can make your own additions here.