**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.

Thu Mar 21 22:22:13 EST 1996