Result 304, Topology

The Hilbert–Smith conjecture in every dimension

Every locally compact second-countable Hausdorff group acting faithfully and jointly continuously on a connected finite-dimensional topological manifold is a Lie group. This proves the Hilbert–Smith conjecture in all finite dimensions, for Hausdorff second-countable manifolds without boundary.

Proof

The bigger picture

Why it matters

Can a space have continuous symmetries too irregular to belong to a smooth mathematical family? The manuscript claims that, under precise topological assumptions, finite-dimensional manifolds cannot have such symmetries when every group element acts distinctly.

What changes?

The manuscript reports this for every finite dimension: a locally compact, second-countable Hausdorff group acting faithfully and jointly continuously on a connected, Hausdorff, second-countable topological manifold without boundary must be a Lie group. A topological manifold is a space locally resembling Euclidean space. Faithfulness means only the identity fixes every point; joint continuity means the action varies continuously with both the group element and the point. The conclusion gives the group a compatible finite-dimensional smooth structure.

What does that help mathematicians do?

The claimed theorem supplies an obstruction to possible symmetry groups. If a group meets the stated topological conditions but is not a Lie group, any jointly continuous action on such a manifold must identify distinct group elements as the same transformation. Researchers could therefore rule out faithful actions without classifying them individually. The conclusion concerns the group's structure, not differentiability of its action.

Are there practical applications?

Its immediate value is foundational: it would connect symmetries defined using continuity alone to the more structured theory of Lie groups. This would constrain which groups can faithfully describe a manifold's symmetries across all finite dimensions. The supplied abstract establishes no computational method or practical deployment; the stated advance is a structural classification in topology.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

The Hilbert–Smith conjecture in every finite dimension

September 23, 2026 46 pages

We prove the Hilbert–Smith conjecture in every finite dimension: every locally compact second-countable Hausdorff group acting faithfully and jointly continuously on a connected Hausdorff second-countable finite-dimensional topological manifold without boundary is a Lie group.

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Cite (BibTeX) 
@misc{OAI:The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026,
  author = {{OpenAI}},
  title = {{The Hilbert--Smith conjecture in every finite dimension}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026/paper.pdf}{OAI:The-Hilbert-Smith-conjecture-in-every-finite-dimension-September-23-2026}},
  year = {2026}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 9 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.