Moderately Discontinuous Homology
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We introduce a new metric homology theory, which we call Moderately Discontinuous Homology, designed to capture Lipschitz properties of metric singular subanalytic germs. The main novelty of our approach is to allow “moderately discontinuous” chains, which are specially advantageous for capturing the subtleties of the outer metric phenomena. Our invariant is a finitely generated graded abelian group (Formula presented.) for any (Formula presented.) and homomorphisms (Formula presented.) for any (Formula presented.). Here (Formula presented.) is a “discontinuity rate”. The homology groups of a subanalytic germ with the inner or outer metric are proved to be finitely generated and only finitely many homomorphisms (Formula presented.) are essential. For (Formula presented.) Moderately Discontinuous Homology recovers the homology of the tangent cone for the outer metric and of the Gromov tangent cone for the inner one. In general, for (Formula presented.) -Homology recovers the homology of the punctured germ. Hence, our invariant can be seen as an algebraic invariant interpolating from the germ to its tangent cone. Our homology theory is a bi-Lipschitz subanalytic 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; this makes the theory quite flexible. In the complex analytic setting 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, and 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. © 2020 Wiley Periodicals LLC.