Lipschitz Normally Embedding and Moderately Discontinuous Homology

Xuan Viet Nhan Nguyen
Basque Center for Applied Mathematics (BCAM), Spain
https://sites.google.com/site/nguyenxvnhan/home

Date(s) : 11/03/2021   iCal
14 h 00 min - 15 h 00 min

In [1] J. Bobadilla et al introduced a homology called Moderately Discontinuous homology (MD-homology) in order to capture the homology of a given germ after collapsing with certain speed. A subanalytic germ $(X, 0)$ is called LNE (Lipschitz normally embedded) if the inner metric and the outer metric on $(X,0)$ are bi-Lipschitz equivalent. The identity map on $(X,0)$ induces homomorphisms between groups of MD-homologies of $(X,0)$ with respect to these two metrics. It is easy to check that if $(X,0)$ is LNE then these homomorphisms are isomorphic. It is asked in the paper that suppose the homomorphisms induced by the identity map are isomorphisms at every point on $(X,0)$, is $(X,0)$ LNE? We will present an example showing that in general, the answer is negative.

[1] J. Bobadilla, S. Heinze, M. Pe Pereira, and J.E. Sampaio, Moderately discontinuous homology, (2020), https://arxiv.org/abs/1910.12552 (preprint).

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