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 for any and homomorphisms for any . Here 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 are essential. For 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 -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.

中文翻译：

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中度不连续同源性

我们引入了一种新的度量同调理论，我们称之为中等不连续同调，旨在捕捉度量奇异子分析胚的 Lipschitz 特性。我们方法的主要新颖之处在于允许“适度不连续”的链，这对于捕捉外部度量现象的微妙之处特别有利。我们的不变量是对 any和对 any的同态的有限生成分级阿贝尔群。这是一个“不连续率”。具有内部或外部度量的子解析胚的同调群被证明是有限生成的，并且只有有限多个同态是必要的。为了中等不连续同源性恢复了外部度量的切锥的同源性和内部度量的 Gromov 切锥的同源性。通常，for -Homology 恢复了被刺破的胚芽的同源性。因此，我们的不变量可以看作是从胚芽到其切锥的代数不变量。我们的同调理论是双李普希茨亚解析不变量，通过适当的度量同伦不变量，并且满足相对和 Mayer-Vietoris 长精确序列的版本。此外，固定不连续率b我们证明它对于一类不连续 Lipschitz 映射是泛函的，其不连续性是b- 适度的；这使得该理论非常灵活。在复杂的分析设置中，我们引入了一种称为 Framed MD 同源性的增强功能，它考虑了来自基本类的信息。作为应用，我们证明了中等不连续同源性表征了所有复杂分析胚中的光滑胚，并恢复了复杂分析胚的不可约分量的数量和平面分支的嵌入拓扑类型。Framed MD同源性恢复了任何平面曲线奇点的拓扑类型和复杂分析胚的相对多重性。© 2020 威利期刊有限责任公司。