Cellular Spaces, Null Spaces and Homotopy Localization

This work offers an exposition on development in homotopy theory. It relates to advances in periodicity in homotopy localization and in cellular spaces. The notion of homotopy localization is treated generally, and encompasses all the known idempotent homotopy functors. It is applied to K-theory localization, to Morava-theories, and to Hopkins-Smith theory of types. The method of homotopy colimits is used heavily.