Moderately discontinuous homology of real surfaces
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2024
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| author | Medeiros, Davi Lopes Sampaio, José Edson Souza, Emanoel |
| author_facet | Medeiros, Davi Lopes Sampaio, José Edson Souza, Emanoel |
| contents | The Moderately Discontinuous Homology (MD-Homology, for short) was created recently in 2022 by Fernández 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00851 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Moderately discontinuous homology of real surfaces Medeiros, Davi Lopes Sampaio, José Edson Souza, Emanoel Metric Geometry Algebraic Geometry Algebraic Topology 51F30, 14B05, 14P10, 55N35 The Moderately Discontinuous Homology (MD-Homology, for short) was created recently in 2022 by Fernández 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. |
| title | Moderately discontinuous homology of real surfaces |
| topic | Metric Geometry Algebraic Geometry Algebraic Topology 51F30, 14B05, 14P10, 55N35 |
| url | https://arxiv.org/abs/2408.00851 |