Quantifying discontinuity
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866914290776145920 |
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| author | Adams, Henry Frick, Florian Harrison, Michael Sadovek, Nikola Superdock, Matt |
| author_facet | Adams, Henry Frick, Florian Harrison, Michael Sadovek, Nikola Superdock, Matt |
| contents | Given a compact space $X$ that does not admit an embedding (an injective continuous function) into $\mathbb{R}^d$, we study the ''degree'' of discontinuity that any injective function $X \to \mathbb{R}^d$ must have. To this end, we define a scale invariant modulus of discontinuity and obtain general lower bounds, thus obtaining quantified nonembeddability results of Haefliger--Weber type. Moreover, we establish analogous lower bounds for simplicial complexes that do not admit an almost $r$-embedding in $\mathbb{R}^d$, thus obtaining a quantified version of the topological Tverberg theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_07636 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantifying discontinuity Adams, Henry Frick, Florian Harrison, Michael Sadovek, Nikola Superdock, Matt Metric Geometry Geometric Topology 51F30, 57N35, 52A35 Given a compact space $X$ that does not admit an embedding (an injective continuous function) into $\mathbb{R}^d$, we study the ''degree'' of discontinuity that any injective function $X \to \mathbb{R}^d$ must have. To this end, we define a scale invariant modulus of discontinuity and obtain general lower bounds, thus obtaining quantified nonembeddability results of Haefliger--Weber type. Moreover, we establish analogous lower bounds for simplicial complexes that do not admit an almost $r$-embedding in $\mathbb{R}^d$, thus obtaining a quantified version of the topological Tverberg theorem. |
| title | Quantifying discontinuity |
| topic | Metric Geometry Geometric Topology 51F30, 57N35, 52A35 |
| url | https://arxiv.org/abs/2511.07636 |