Davi Lopes Medeiros, José Edson Sampaio, Emanoel Souza
{"title":"Moderately discontinuous homology of real surfaces","authors":"Davi Lopes Medeiros, José Edson Sampaio, Emanoel Souza","doi":"arxiv-2408.00851","DOIUrl":null,"url":null,"abstract":"The Moderately Discontinuous Homology (MD-Homology, for short) was created\nrecently in 2022 by Fern\\'andez de Bobadilla at al. and it captures deep\nLipschitz phenomena. However, to become a definitive powerful tool, it must be\nwidely comprehended. In this paper, we investigate the MD-Homology of definable surface germs for\nthe inner and outer metrics. We completely determine the MD-Homology of\nsurfaces for the inner metric and we present a great variety of interesting\nMD-Homology of surfaces for the outer metric, for instance, we determine the\nMD-Homology of some bubbles, snake surfaces, and horns. Furthermore, we\nexplicit the diversity of MD-Homology of surfaces for the outer metric in\ngeneral, showing how hard it is to completely solve the outer classification\nproblem. On the other hand, we show that, under specific conditions, the weakly\nouter Lipschitz equivalence determines completely the MD-Homology of surfaces\nfor the outer metric, showing that these two subjects are quite related.","PeriodicalId":501444,"journal":{"name":"arXiv - MATH - Metric Geometry","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Metric Geometry","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2408.00851","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
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.