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The classical Borsuk-Ulam theorem states that any continuous map $ f: S^n \to \mathbb{R}^n $ from an $n $-dimensional sphere to $ n $-dimensional Euclidean space has a point $ x \in S^n $ such that $ f(x) = f(-x) $. This theorem has far-reaching consequences in topology, geometry, and combinatorics, and it has been generalized in various ways, including versions involving higher-dimensional spaces and different topological structures.

I am interested in understanding whether a $ p $-adic version of the Borsuk-Ulam theorem exists or could be formulated. Specifically, I would like to ask the following questions:

  1. Existence of $ p $-adic Counterparts:

    • Is there a natural counterpart to the $ n $-dimensional sphere $ S^n $ in the context of $ p $-adic spaces (e.g., $ \mathbb{Q}_p^n $ or analogous spaces)?
    • Can a continuous map $ f: S^n_p \to \mathbb{Q}_p^n $ (or some $p $-adic analogue of $ \mathbb{R}^n $) be defined such that we can expect a result similar to the Borsuk-Ulam theorem, where $f(x) = f(-x) $?

  2. Symmetry in $ p $-adic Spaces:

    • How does the non-Archimedean nature of the $ p $-adic metric affect the formulation of symmetry results like the Borsuk-Ulam theorem?
    • If such a result is possible, what structural properties or constraints of $ p $-adic spaces would be necessary for the existence of symmetric points (i.e., points $ x $ such that $f(x) = f(-x) $)?

  3. Known Results or Obstructions:

    • Are there any known theorems or results in $ p $-adic topology or geometry that deal with symmetric points or mappings analogous to the Borsuk-Ulam theorem?
    • If no such result exists, are there known counterexamples or reasons why the Borsuk-Ulam theorem does not have an analog in the $ p $-adic setting?

  4. Comparisons Between $\mathbb{R}^n $ and $ \mathbb{Q}_p^n $:

    • How can we compare $\mathbb{R}^n$ with $ \mathbb{Q}_p^n $ in terms of their topological properties, and how does this comparison influence the possibility of generalizing the Borsuk-Ulam theorem to the $ p $-adic case?
    • Is there any research connecting these two spaces, especially regarding symmetries or the behavior of continuous maps between them?

I would appreciate any insights, references, or pointers to related work on whether a $ p $-adic version of the Borsuk-Ulam theorem could be formulated, or if there are known obstacles in defining such an extension.

Note: I have also posted a version of the same question in MSE community, here is the cross-posting link.

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    $\begingroup$ On Mathematics: Is There a $p$-adic Analog of the Borsuk-Ulam Theorem? $\endgroup$
    – Martin Sleziak
    Commented 6 hours ago
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    $\begingroup$ The fact $\mathbb Q_p^n$ is totally disconnected is going to be a severe obstruction to any such result. You have continuous functions $\mathbb Q_p^n\setminus\{0\}\to\{0,1\}$ such that for every $x$, we have $f(-x)\neq f(x)$. $\endgroup$
    – Wojowu
    Commented 3 hours ago
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    $\begingroup$ You shouldn't cross-post questions nearly simultaneously. If you absolutely have to, let a question sit on math.SE for a week at least, before bringing it over to MO. $\endgroup$
    – David Roberts ♦
    Commented 2 hours ago
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    $\begingroup$ Also, and this applies to your other question too, you should ask one self-contained question in a posting, not eight. And, in addition, asking two very big (barrages of) questions in close succession is asking a lot of the community. There shouldn't be such a hurry that you need both of these at once. $\endgroup$
    – David Roberts ♦
    Commented 2 hours ago

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