>>10506194

Let's think about a simple example, the circle. I can break up the circle into two slightly overlapping pieces, such that if I glue them back together (with the overlap) I get back the circle. We call this a "union" of the two pieces (line segments).

Now let's think about that little bit of overlap: they overlap at the ends of each pieces, so there are two disconnected bits if we only think about those overlaps. We call that the "intersection".

Now, recall that the kth de Rham cohomology of a manifold M is the kernel of the exterior derivative on the k-forms on M mod the image of (k-1)-forms. There is a short exact sequence from the 0 vector space into the kth de Rham cohomology of the circle, into the (outer) direct sum of the kth de Rham cohomologies of each piece of the circle, into the kth de Rham cohomology of their intersections, and then back into 0. We can see this existence intuitively by thinking about cutting apart and the patching back together these particular pieces of a circle: there should be a map that lets us preserve properties on each of those pieces.

The Mayer-Vietoris sequence, then, is the existence of connecting maps that connect each kth short exact sequence of this form with the (k+1)th short exact sequence of the same form. That is, as we move up cohomology degree (in de Rham theory, that is the degree of the exterior forms that describe the geometry of the manifold), we have a cochain of maps which such that we can relate each kth cohomology with the next, and thus compute them all efficiently via this cutting and patching method.