A Mayer-Vietoris Sequence is a powerful tool used in finding Homology groups for spaces that can be expressed as the unions of simpler spaces from the perspective of Homology theory.
If X is a topological space covered by the interiors of two subspaces A and B, then
is an exact sequence where . There is a slight adaptation for the reduced homology where the sequence ends instead
Consider the cover of formed by 2-discs A and B in the figure.
The space is homotopy equivalent to the circle. We know that the homology groups are preserved by homotopy and so for and . Also note how the homology groups of A and B are trivial since they are both contractable. So we know that
This means that since is an isomorphism by exactness.
Consider the cover of the torus by 2 open ended cylinders A and B.