Regular and Semisimplicial CW Complexes

Abstract

In this chapter we will study two special kinds of CW complexes. Regular complexes, the first topic, are interesting because they are very useful in homological calculations. As we shall see in Chapter V, they share with simplicial complexes the virtue of allowing a relatively easy calculation of the “algebraic boundary” of a cell. At the same time, they share with more general CW complexes the advantage that often a space can be represented as a regular CW complex with many fewer cells than in a simplicial decomposition. A regular CW complex can be subdivided into a simplicial complex: in this sense it is a simplicial complex in which the simplexes are more efficiently combined into closed cells. For example, the ⊗ product of any two regular complexes is again a regular complex. If they are simplicial complexes, then the cells of the product complex are subdivided (in a nonunique way) to give the standard product simplicial complex.