A Perturbative Multiplicity Theorem for the Borsuk-Ulam Setting
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908462539079680 |
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| author | Mansour, Karim |
| author_facet | Mansour, Karim |
| contents | We prove a generalization of the classical Borsuk--Ulam Theorem under small perturbations (shaking) of the sphere. We show that for a generic perturbation of a continuous map $f : S^2 \to \mathbb{R}^2$, the number of points $x \in S^2$ such that $f_ε(x) = f_ε(-x)$ becomes finite and odd, and may exceed the classical lower bound of one antipodal coincidence. In particular, we show the existence of maps with 3, 5, or 7 such points, and explain the unbounded nature of this multiplicity under higher complexity of the perturbation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_11556 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Perturbative Multiplicity Theorem for the Borsuk-Ulam Setting Mansour, Karim General Mathematics We prove a generalization of the classical Borsuk--Ulam Theorem under small perturbations (shaking) of the sphere. We show that for a generic perturbation of a continuous map $f : S^2 \to \mathbb{R}^2$, the number of points $x \in S^2$ such that $f_ε(x) = f_ε(-x)$ becomes finite and odd, and may exceed the classical lower bound of one antipodal coincidence. In particular, we show the existence of maps with 3, 5, or 7 such points, and explain the unbounded nature of this multiplicity under higher complexity of the perturbation. |
| title | A Perturbative Multiplicity Theorem for the Borsuk-Ulam Setting |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2507.11556 |