Mayer-Vietoris sequence
Let is a topological space, and are such that , and . Then there is an exact sequence
of homology groups:
Here, is induced by the inclusions and by , and is the following map:if is in , then it can be written as the sum of a chain in and one in , ., since . Thus, is a chain in , and so representsa class in . This is . One can easily check (by standard diagram chasing) that this map is well defined on the level of homology.