Week 9 - Part 2
The Mayer-Vietoris Theorem
Proposition B. (``A short exact sequence of cochain complexes gives a long exact sequence in cohomology.'') Consider an exact sequence of cochain complexes
Then for each p there exists a linear map which, together with the maps in cohomology induced by i and j, fits into the exact sequence:
Proof. Standard diagram-chasing argument. The corresponding theorem for chain complexes and homology groups (formally identical, except the d's go down in dimension instead of up) is Bredon's Theorem 5.6, which is proved in detail.
Proposition C. Suppose a smooth manifold M is the union of two open sets U and V. Let , , , be the inclusions. The following sequence of cochain-complex homomorphisms is exact.
Proof. Exactness at the first two nodes is completely straightforward. Now suppose . In general will not extend to either or separately. Let be a smooth partition of unity subordinate to the cover U, V. Then
gives a smooth p-form on U; similarly 0 and , or more conveniently , define an element . On , , proving exactness at the third node.
The Mayer-Vietoris Theorem follows directly from Propositions B and C.