Function-Rips complexes in persistent homotopy theory: Local stability and Latschev theorems
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866918407079723008 |
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| author | Oudot, Steve Waas, Lukas |
| author_facet | Oudot, Steve Waas, Lukas |
| contents | Latschev's theorem provides sufficient conditions on a metric space $M$ and $δ> 0$ for the homotopy type of $M$ to agree with that of the Vietoris-Rips complex $\mathcal{R}^δ(N)$ of any nearby space $N$ in the Gromov-Hausdorff distance. We prove a persistent version of this theorem, providing sufficient conditions on a pair $(M, f \colon M \to \mathbb{R}^N)$ and $δ> 0$ for the persistent homotopy type of the sublevel set filtration of $(M,f)$ to be interleaved with that of the function-Rips complex $\mathcal{R}^δ(N^{\bullet})$ of any nearby pair $(N,g)$. In particular, our result answers a longstanding question on the related topic of estimating sublevel set persistent homology from finite point samples. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23460 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Function-Rips complexes in persistent homotopy theory: Local stability and Latschev theorems Oudot, Steve Waas, Lukas Algebraic Topology Metric Geometry 55U10, 55Pxx, 54E35, 55N31, 68U05 Latschev's theorem provides sufficient conditions on a metric space $M$ and $δ> 0$ for the homotopy type of $M$ to agree with that of the Vietoris-Rips complex $\mathcal{R}^δ(N)$ of any nearby space $N$ in the Gromov-Hausdorff distance. We prove a persistent version of this theorem, providing sufficient conditions on a pair $(M, f \colon M \to \mathbb{R}^N)$ and $δ> 0$ for the persistent homotopy type of the sublevel set filtration of $(M,f)$ to be interleaved with that of the function-Rips complex $\mathcal{R}^δ(N^{\bullet})$ of any nearby pair $(N,g)$. In particular, our result answers a longstanding question on the related topic of estimating sublevel set persistent homology from finite point samples. |
| title | Function-Rips complexes in persistent homotopy theory: Local stability and Latschev theorems |
| topic | Algebraic Topology Metric Geometry 55U10, 55Pxx, 54E35, 55N31, 68U05 |
| url | https://arxiv.org/abs/2603.23460 |