Metric homology
Lev Birbrair, Jean‐Paul Brasselet · Communications on Pure and Applied Mathematics · 2000
Metric homology is a homology theory constructed on semialgebraic (or compact subanalytic) sets with singularities. Metric homology is an invariant under semialgebraic (subanalytic) bi-Lipschitz homeomorphisms and not a topological invariant. As in intersection homology theory, classes of admissible chains are defined using a semialgebraic stratification and a perversity function. In contrast to intersection homology, the perversity is a rational-valued function. Instead of the topological dimension of the intersection of a chain with a stratum, we consider the so-called volume-growth number. This number is a sort of generalization of Hausdorff dimension. In the second part of the paper we describe one-dimensional metric homology for spaces with isolated singularities and calculate some concrete examples. © 2000 John Wiley & Sons, Inc.