1-Dimensional Intrinsic Persistence of Geodesic Spaces
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2017
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| _version_ | 1866913502358142976 |
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| author | Virk, Žiga |
| author_facet | Virk, Žiga |
| contents | Given a compact geodesic space $X$ we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or Čech filtration of $X$ to obtain what we call a persistence. This paper contains the theory describing such persistence: properties of the set of critical points, their precise relationship to the size of holes, the structure of persistence and the relationship between open and close, Rips and Čech induced persistences. Amongst other results we prove that a Rips critical point $c$ corresponds to an isometrically embedded circle of length $3c$, that a homology persistence of a locally contractible space with coefficients in a field encodes the lengths of the lexicographically smallest base and that Rips and Čech induced persistences are isomorphic up to a factor $3/4$. The theory describes geometric properties of the underlying space encoded and extractable from persistence. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1709_05164 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | 1-Dimensional Intrinsic Persistence of Geodesic Spaces Virk, Žiga Geometric Topology Algebraic Topology Metric Geometry Given a compact geodesic space $X$ we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or Čech filtration of $X$ to obtain what we call a persistence. This paper contains the theory describing such persistence: properties of the set of critical points, their precise relationship to the size of holes, the structure of persistence and the relationship between open and close, Rips and Čech induced persistences. Amongst other results we prove that a Rips critical point $c$ corresponds to an isometrically embedded circle of length $3c$, that a homology persistence of a locally contractible space with coefficients in a field encodes the lengths of the lexicographically smallest base and that Rips and Čech induced persistences are isomorphic up to a factor $3/4$. The theory describes geometric properties of the underlying space encoded and extractable from persistence. |
| title | 1-Dimensional Intrinsic Persistence of Geodesic Spaces |
| topic | Geometric Topology Algebraic Topology Metric Geometry |
| url | https://arxiv.org/abs/1709.05164 |