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.