Module 6 : Basic homology theory
Lecture 35 :The Mayer Vietoris sequence and its applications
 

The proof of Mayer Vietoris sequence is reminiscent of the Seifert Van Kampen theorem. While the Seifert Van Kampen theorem enables us to relate the fundamental group of a union $ U\cup V$ in terms of the fundamental groups of $ U, V$ ,and $ U\cap V$, the situation here is slightly more involved. The precise relationship between the homologies of $ U, V$and $ U\cup V$ is described in terms of the long exact sequence of theorem (34.7).

As in the Seifert Van Kampen theorem we obtain from a push-out diagram of topological spaces a push out diagram of chain complexes which turns into a short exact sequence of complexes. The corresponding long-exact sequence gives, after an application of the excision theorem of the last lecture, the Mayer Vietoris sequence. It is one of the most efficient tools available for the computation of homology groups. We restate here the theorem for convenience.

Theorem 35.1:
Suppose $ U$ and $ V$ are subsets of a topological space such that Int   Int$ \; U\cup$. Then there is a long exact sequence