Homology of Maps

Abstract

In Chapter 5 we have provided a preview of the issues involved in defining homology of maps. In this chapter we revisit this material, but in a rigorous and dimension-independent manner. We begin with the introduction of representable sets. These are sets that can be constructed using elementary cells and represent a larger class than that of cubical sets. This extra flexibility is used in Section 6.2 to construct cubical multivalued maps. As described in Section 5.2, these multivalued maps provide representations of the continuous function for which we wish to compute the homology map. Section 6.3 describes the process by which one passes from the cubical map to a chain map from which one can define a map on homology. Section 6.4 shows that applying the above-mentioned steps (plus perhaps rescaling) to a continuous function leads to a well-defined map on homology. Finally, in the last section, we address the question of when do two different continuous maps give rise to the same map on homology.