论文标题
中度不连续的同副本
Moderately Discontinuous Homotopy
论文作者
论文摘要
我们介绍了一种度量同型理论，我们称之为中度不连续的同质镜，旨在捕获度量单数亚分析细菌的Lipschitz性质。它与作者和E. sampaio的中等不连续的同源理论相匹配。 $ k $ -th的MD同型组是一个组$ mdh^b_ \ bullet $，用于[1，\ infty] $中的任何$ b \ for In [1，\ infty] $中的任何$ b \ geq b \ geq b'$。我们开发了所有基本特性，包括对组的有限表示，对长的同源序列，度量同型不变性，Seifert-Van Kampen定理和Hurewicz同构定理。我们证明了比较定理，允许将度量同质组与相关空间的拓扑同喻组联系起来。对于$ b = 1 $，它恢复了外部度量标准的切线锥和内部指标的同型组。通常，对于$ b = \ infty $，$ md $ - homotophy恢复了刺穿细菌的同质。因此，我们的不变式可以看作是代数不变的，将从细菌到切线锥的同质插值。我们以对内部度量的任何正常表面奇异性的全面计算结束纸张。在同一情况下，我们还提供了MD-词素的完整计算。
We introduce a metric homotopy theory, which we call Moderately Discontinuous Homotopy, designed to capture Lipschitz properties of metric singular subanalytic germs. It matches with the Moderately Discontinuous Homology theory receantly developed by the authors and E. Sampaio. The $k$-th MD homotopy group is a group $MDH^b_\bullet$ for any $b\in [1,\infty]$ together with homomorphisms $MDπ^b\to MDπ^{b'}$ for any $b\geq b'$. We develop all its basic properties including finite presentation of the groups, long homology sequences of pairs, metric homotopy invariance, Seifert-van Kampen Theorem and the Hurewicz isomorphism Theorem. We prove comparison theorems that allow to relate the metric homotopy groups with topological homotopy groups of associated spaces. For $b=1$ it recovers the homotopy groups of the tangent cone for the outer metric and of the Gromov tangent cone for the inner one. In general, for $b=\infty$ the $MD$-homotopy recovers the homotopy of the punctured germ. Hence, our invariant can be seen as an algebraic invariant interpolating the homotopy from the germ to its tangent cone. We end the paper with a full computation of our invariant for any normal surface singularity for the inner metric. We also provide a full computation of the MD-Homology in the same case.
