Moderately Discontinuous Homology
We introduce a new metric homology theory, Moderately Discontinuous Homology, which captures Lipschitz properties of metric subanalytic germs. The main novelty 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 $MDH^b_\bullet$ for any $b\in [1,\infty]$ and homomorphisms $MDH^b_\bullet\to MDH^{b'}_\bullet$ for any $b\geq b'$. Here $b$ is a "discontinuity rate". The homology groups for the inner or outer metric are proved to be finitely generated and that only finitely many homomorphisms $MDH^b_\bullet\to MDH^{b'}_\bullet$ are essential. For $b=1$ it recovers the homology of the tangent cone for the outer metric and of the Gromov tangent cone for the inner one. In general, for $b=\infty$ the $MD$- homology recovers the homology of the punctured germ. Hence, our invariant interpolates from the germ to its 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; 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, 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.