Metric homology (Q2711841)

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 1-dimensional metric homology for spaces with isolated singularities and calculate some concrete examples.