Alexander's duality theorem

[摘要] Topology or Analysis Situs is usually defined as the study of properties of spaces or their configurations under continuous transformations. The invariants under these transformations play a leading role in the study of the subject, and it is with them that the duality theorems are concerned. The first duality theorem was the theorem of Poincare on the duality of the Betti numbers of orientable manifolds. This theorem is stated in terms of dual complexes and is quite different from the duality theorem of Alexander which concerns residual spaces. The latter theorem may also be stated in terms of Betti numbers, but it was proved for a different set of invariants, the connectivity numbers, by Alexander. The relation between the two kinds of invariants will be shown below. The terminology and notation of Topology vary from one author to another. The definitions of such things as chains and circuits are not the same as given by different authorities. Wherever possible the notations and terminology of Veblen and Alexander have been followed.

[发布机构] Rice University