Finiteness conditions for CW complexes. Il

Abstract

A CW complex is a topological space which is built up in an inductive way by a process of attaching cells. Spaces homotopy equivalent to CW complexes play a fundamental role in topology. In the previous paper with the same title we gave criteria (in terms of more-or-less standard invariants of the space) for a CW complex to be homotopy equivalent to one of finite dimension, or to one with a finite number of cells in each dimension, or to a finite complex. This paper contains some simplification of these results. In addition, algebraic machinery is developed which provides a rough classification of CW complexes homotopy equivalent to a given one (the existence clause of the classification is the interesting one). The results would take a particularly simple form if a certain (rather implausible) conjecture could be established.