Moderately Discontinuous Homology.

In this paper we introduce a new metric homology theory, which we call Moderately Discontinuous Homology. The basic idea involves the following basic ideas: first we define a singular homology theory whose simplexes are families of singular chains depending on a parameter t in (0,1), so that the distance from the support of the chain to the singularity approaches the singularity at speed 1 with respect to the parameter. Second, we allow b-moderately discontinuous chains for a certain discontinuity rate b in ranging from 1 to infinity. Combining the homology group obtained for the different discontinuity rates, we obtain an algebraic invariant that is given by a graded abelian group for any b and homomorphisms between these groups. We prove finitely generation, for b equal to infinity it recovers the homology of the punctured germ, and for b=1 it recovers the homology of the tangent cone. Our homology theory is a bi-Lipschitz subanalitic invariant, is invariant by suitable metric homotopies, and satisfies versions of the relative and Mayer-Vietoris long exact sequences. Moreover, fixed a discontinuity rate b we show that it is functorial for a class of discontinuous Lipschitz maps, whose discontinuities are b-moderated. We introduce an enhancement called Framed MD Homology, which takes into account information from fundamental classes. As applications we prove that Moderately Discontinuous Homology characterizes smooth germs among all complex analytic germs, recovers the number of irreducible components of complex analytic germs and the embedded topological type of plane branches. Framed MD Homology recovers the topological type of any plane curve singularity and relative multiplicities of complex analytic germs.