# Lipschitz Normally Embedding and Moderately Discontinuous Homology

Xuan Viet Nhan Nguyen
Basque Center for Applied Mathematics (BCAM), Spain

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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