MAT 531 Spring 1996 - Topology/Geometry II

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 tex2html_wrap_inline48 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 tex2html_wrap_inline62 , tex2html_wrap_inline64 , tex2html_wrap_inline66 , tex2html_wrap_inline68 be the inclusions. The following sequence of cochain-complex homomorphisms is exact.


Proof. Exactness at the first two nodes is completely straightforward. Now suppose tex2html_wrap_inline72 . In general tex2html_wrap_inline74 will not extend to either tex2html_wrap_inline76 or tex2html_wrap_inline78 separately. Let tex2html_wrap_inline80 be a smooth partition of unity subordinate to the cover U, V. Then


gives a smooth p-form on U; similarly 0 and tex2html_wrap_inline92 , or more conveniently tex2html_wrap_inline94 , define an element tex2html_wrap_inline96 . On tex2html_wrap_inline98 , tex2html_wrap_inline100 , proving exactness at the third node.

The Mayer-Vietoris Theorem follows directly from Propositions B and C.

Tony Phillips
Thu Mar 21 22:22:13 EST 1996