Asymptotic topology

In this paper we study the similarity between local topology and its global analogue, so-called asymptotic topology. In the asymptotic case, the notions of dimension, cohomological dimension, and absolute extensor are introduced and some basic facts about them are proved. The Higson corona functor establishing a connection between macro- and micro-topology is considered. A relationship between problems of general asymptotic topology and the Novikov conjecture on higher signatures is discussed. Some new results concerning the Novikov conjecture and other related conjectures are presented.