Magnitude homology equivalence of Euclidean sets
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911465478291456 |
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| author | Mateo, Adrián Doña Leinster, Tom |
| author_facet | Mateo, Adrián Doña Leinster, Tom |
| contents | Magnitude homology is an $\mathbf{R}^+$-graded homology theory of metric spaces that captures information on the complexity of geodesics. Here we address the question: when are two metric spaces magnitude homology equivalent, in the sense that there exist back-and-forth maps inducing mutually inverse maps in homology? We give a concrete geometric necessary and sufficient condition in the case of closed Euclidean sets. Along the way, we introduce the convex-geometric concepts of inner boundary and core, and prove a strengthening for closed convex sets of the classical theorem of Carathéodory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_11722 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Magnitude homology equivalence of Euclidean sets Mateo, Adrián Doña Leinster, Tom Metric Geometry Algebraic Topology Category Theory 18G90 (Primary) 51F99, 52A99 (Secondary) Magnitude homology is an $\mathbf{R}^+$-graded homology theory of metric spaces that captures information on the complexity of geodesics. Here we address the question: when are two metric spaces magnitude homology equivalent, in the sense that there exist back-and-forth maps inducing mutually inverse maps in homology? We give a concrete geometric necessary and sufficient condition in the case of closed Euclidean sets. Along the way, we introduce the convex-geometric concepts of inner boundary and core, and prove a strengthening for closed convex sets of the classical theorem of Carathéodory. |
| title | Magnitude homology equivalence of Euclidean sets |
| topic | Metric Geometry Algebraic Topology Category Theory 18G90 (Primary) 51F99, 52A99 (Secondary) |
| url | https://arxiv.org/abs/2406.11722 |