|
In mathematics, cohomology with compact support refers to certain cohomology theories, usually with some condition requiring that cocycles should have compact support. ==de Rham cohomology with compact support for smooth manifolds== Given a manifold ''X'', let be the real vector space of ''k''-forms on ''X'' with compact support, and ''d'' be the standard exterior derivative. Then the de Rham cohomology groups with compact support are the homology of the chain complex : : ''i.e.'', is the vector space of closed ''q''-forms modulo that of exact ''q''-forms. Despite their definition as the homology of an ascending complex, the de Rham groups with compact support demonstrate covariant behavior; for example, given the inclusion mapping ''j'' for an open set ''U'' of ''X'', extension of forms on ''U'' to ''X'' (by defining them to be 0 on ''X''–''U'') is a map inducing a map :. They also demonstrate contravariant behavior with respect to proper maps - that is, maps such that the inverse image of every compact set is compact. Let ''f'': ''Y'' → ''X'' be such a map; then the pullback : induces a map :. If ''Z'' is a submanifold of ''X'' and ''U'' = ''X''–''Z'' is the complementary open set, there is a long exact sequence : called the long exact sequence of cohomology with compact support. It has numerous applications, such as the Jordan curve theorem, which is obtained for ''X'' = R² and ''Z'' a simple closed curve in ''X''. De Rham cohomology with compact support satisfies a covariant Mayer–Vietoris sequence: if ''U'' and ''V'' are open sets covering ''X'', then : where all maps are induced by extension by zero is also exact. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「cohomology with compact support」の詳細全文を読む スポンサード リンク
|