Density and unique decomposition theorems for the lattice of cellular classes

Abstract

A class C of pointed spaces is called a cellular class if it is closed under weak equivalences, arbitrary wedges and pointed homotopy pushouts. The smallest cellular class containing X is denoted by C(X), and a partial order relation much less than is defined by: X much less than Y if Y is an element of C(X). In this text we investigate the sub partial order sets generated respectively by simply connected finite CW-complexes and by rational spaces. For rational spaces we prove a unique decomposition theorem, a density theorem and the existence of infinitely many non-comparable elements. We then prove the density theorem for a generic class of finite CW-complexes.