Moderately discontinuous homology of real surfaces

TL;DR

This paper studies the Moderately Discontinuous Homology (MD-Homology) of real surfaces under different metrics, providing complete results for inner metrics and diverse examples for outer metrics, highlighting the complexity of classification.

Contribution

It offers a complete determination of MD-Homology for surfaces with inner metrics and explores the diversity and classification challenges for outer metrics.

Findings

Complete MD-Homology for surfaces with inner metric.

Diverse MD-Homology examples for outer metric surfaces.

Weak outer Lipschitz equivalence determines MD-Homology under certain conditions.

Abstract

The Moderately Discontinuous Homology (MD-Homology, for short) was created recently in 2022 by Fern\'andez de Bobadilla at al. and it captures deep Lipschitz phenomena. However, to become a definitive powerful tool, it must be widely comprehended. In this paper, we investigate the MD-Homology of definable surface germs for the inner and outer metrics. We completely determine the MD-Homology of surfaces for the inner metric and we present a great variety of interesting MD-Homology of surfaces for the outer metric, for instance, we determine the MD-Homology of some bubbles, snake surfaces, and horns. Furthermore, we explicit the diversity of MD-Homology of surfaces for the outer metric in general, showing how hard it is to completely solve the outer classification problem. On the other hand, we show that, under specific conditions, the weakly outer Lipschitz equivalence determines completely the MD-Homology of surfaces for the outer metric, showing that these two subjects are quite related.