{"paper":{"title":"The Hilbert--Smith conjecture for three-manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GT","authors_text":"John Pardon","submitted_at":"2011-12-11T06:24:54Z","abstract_excerpt":"We show that every locally compact group which acts faithfully on a connected three-manifold is a Lie group. By known reductions, it suffices to show that there is no faithful action of $\\mathbb Z_p$ (the $p$-adic integers) on a connected three-manifold. If $\\mathbb Z_p$ acts faithfully on $M^3$, we find an interesting $\\mathbb Z_p$-invariant open set $U\\subseteq M$ with $H_2(U)=\\mathbb Z$ and analyze the incompressible surfaces in $U$ representing a generator of $H_2(U)$. It turns out that there must be one such incompressible surface, say $F$, whose isotopy class is fixed by $\\mathbb Z_p$. A"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.2324","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}