Ultrametrics on Čech homology groups

Abstract

To the memory of Sibe Mardesič to whom we will always be indebted MSC: 55N05 55P55 Keywords:Čech homology Čech homology groups Approximative cycle and boundary Approximative homology Ultrametric This paper is devoted to introducing additional structure on Čech homology groups. First, we redefine the Čech homology groups in terms of what we have called approximative homology by using approximative sequences of cycles, just as Borsuk introduced shape groups using approximative maps. From this point on, we are able to construct complete ultrametrics on Čech homology groups. The uniform type (and then the group topology) generated by the ultrametric leads to a shape invariant which we use to deduce topological consequences.