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Integral tail signatures and the Hilbert--Smith conjecture

Primarily AI-generated textHuman understanding: some partsmath.GT — Geometric Topologymath.AT — Algebraic Topologymath.GR — Group Theorymath.KT — K-Theory and Homology

Contributed by Alexander Dabrowski ↗

Version 1 / Oct 16, 2026 / CC BY 4.0

Abstract

The obstruction to an effective pp-adic manifold action is an integer signature tail. On asymptotically controlled rational modules over one fixed free CpC_p-space, finite-cover Mayer--Vietoris makes the difference between pp copies and quotient-permutation tensoring nilpotent. Every degree-four class therefore has eventually pp-divisible signatures. An effective action gives a class of signature 11: coordinate averaging provides degree-one sphere control, free sheets turn quotient representatives into a compatible homotopy idempotent, and its normalized Poincar\'e corner forgets to a scalar manifold germ. Successive localization boundaries recover one copy of $\CP^2$, up to sign. The resulting contradiction proves the Hilbert--Smith conjecture.

Provenance statement

I checked background, references, abstract, introduction, soundness in depth. I went through the whole paper as in depth as I could and understand all the main ideas, but do not fully understand all the technical details of all the proofs. However multiple LLMs (ChatGPT Pro, Claude Opus 5.5/Fable 5.1) audited it extensively. Moreover the linked Lean 4 formalization verifies the main results under second countability assumptions.

Formalizations

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Version history

  1. v1Submitted by Alexander DabrowskiInitial depositCurrentOct 16, 2026