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Math-2207 · mathematics · Topology

Kirby Problem 5.3

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

The Hilbert--Smith assertion asks whether every locally compact group acting faithfully and continuously on a connected manifold is a Lie group. The companion question concerns an action of a Cantor group on a Euclidean neighborhood retract: must the subset $A$ of points with trivial stabilizer be a homology-$Z$-subset? This homological negligibility means that $H_*(U,U\setminus A;\mathbb Z)=0$ for every open subset $U$ of the ambient neighborhood retract.

Short English statement

The Hilbert--Smith assertion asks whether every locally compact group acting faithfully and continuously on a connected manifold is a Lie group. The companion question concerns an action of a Cantor group on a Euclidean neighborhood retract: must the subset $A$ of points with trivial stabilizer be a homology-$Z$-subset? This homological negligibility means that $H_*(U,U\setminus A;\mathbb Z)=0$ for every open subset $U$ of the ambient neighborhood retract.

Sources

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2207-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2207

Manuscript 1 (family 304, 2026-09-23): The Hilbert–Smith bullet is proved in every finite manifold dimension, but the second bullet asks the stronger homology-Z-set assertion for Cantor-group actions on arbitrary ENRs, which is not proved.

Kirby Problem 5.3

OpenAI's claim

A faithful continuous action of a second-countable locally compact group on any connected finite-dimensional topological manifold is claimed to force the group to be Lie, resolving Hilbert--Smith in every finite dimension. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For every finite integer $n\ge1$, OpenAI asserts that a Hausdorff second-countable group $G$, locally compact, with a faithful action on a connected topological $n$-manifold $M$. Require $M$ to be Hausdorff, second-countable and without boundary, and the action map $G\times M\to M$ to be continuous. Under these conditions $G$ is a Lie group. Its $p$-adic exclusion theorem says \[ \ker(\mathbb Z_p\curvearrowright M)\ne\{0\},\qquad \mathbb Z_p=\varprojlim_k\mathbb Z/p^k\mathbb Z, \] for every prime $p$ and every such jointly continuous action; therefore some finite quotient already determines the action. Faithfulness means trivial global kernel, and does not mean free action.

Outline of the argument

Manuscript 1 (family 304, 2026-09-23): Witt groups of sheaves generated by continuous polyhedral maps acquire a fixed integral signature lattice through cutoff localization and doubling comparisons with semialgebraic objects. A hypothetical p-adic action produces equally valued classes in arbitrarily large orbit batches. Their rational values would have unbounded p-power denominators, contradicting the lattice fixed before the batch size is chosen.

Formal verification

Family 304 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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