Moderately Discontinuous Homology

Abstract

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 MDH b for any b 2 1; C1 and homomorphisms MDH b ! MDH b 0 for any b b 0 . Here b 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 MDH b ! MDH b 0 are essential. For b D 1 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 b D C1 the MD-Homology recovers the homologyof 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.

On a link criterion for Lipschitz normal embeddings among definable sets

“…This section presents a counterexample to a question asked in [6] about a sufficient condition on MD‐homologies for a subanalytic germ to be Lipschitz normally embedded (see Question 3.1). MD‐homology is defined in a similar way to singular homology but quotienting a b ‐proximity relation.…”

“…In [6], Bobadilla–Heinze–Pereira–Sampaio introduced a homology theory called moderately discontinuous homology (MD‐homology) in order to capture the singular homology of the link of a given subanalytic germ ( X , 0) after collapsing it at a certain speed. The identity map $${\rm Id }_{(X,0)}: (X, 0, d_{\text{inn}}) \rightarrow (X,0, d_{\text{out}})$$ induces homomorphisms between groups of MD‐homologies of $$(X,x_0)$$ for all $$x_0 \in (X,0)$$.…”

“…It is easy to check that if ( X , 0) is LNE then these homomorphisms are actually isomorphisms. It is asked in [6] if the converse holds, that is, suppose these homomorphisms are isomorphisms then is it true that ( X , 0) is LNE. In Section 3, as an application of Theorem 1.2 (also Theorem 1.5) we give a simple example showing that in general the answer is negative.…”

On a link criterion for Lipschitz normal embeddings among definable sets

“…We briefly recall the definition of Moderately discontinuous homology. For more details about this theory, we refer the reader to [5]. Definition 3.1.…”

“…In [5] Bobadilla-Heinze-Pereira-Sampaio introduced a homology called Moderately Discontinuous homology (MD homology) in order to capture the singular homology of the link of given subanalytic germ (X, 0) after collapsing with a certain speed. The identity map Id : (X, 0, d inn ) → (X, 0, d out ) induces homomorphisms between groups of MD-homologies of (X, x 0 ) for all x 0 ∈ (X, 0).…”