Introduction

Abstract

This chapter introduces homotopy category NH of nilpotent CW-complexes. Here a pointed space X is said to be nilpotent if its fundamental group is a nilpotent group and operates nilpotently on the higher homotopy groups. For a given family P of rational primes, the concept of a P-local space is based simply on the requirement that its homotopy groups be P-local. Thus a localization theory for the category NH requires, or involves, a localization theory of nilpotent groups and of nilpotent actions of nilpotent groups on abelian groups. This latter theory could be obtained as a by-product of the topological theory. The chapter presents the exposition of the theory of localization of nilpotent groups and homotopy types. A study of the localization theory of nilpotent groups and nilpotent actions is presented.