Main theorems
The goal of the book: a locally compact group acting faithfully and continuously on a connected manifold must be a Lie group. The paper reduces this to excluding faithful actions of the \(p\)-adic integers, proves that exclusion by contradiction on an invariant chart subset, and records a dynamical consequence for almost-periodic homeomorphisms.
The statements
Let \(n \geq 1\). A locally compact second-countable Hausdorff group acting faithfully and jointly continuously on a connected Hausdorff second-countable topological \(n\)-manifold without boundary is a Lie group.
Proof sketch. The classical Hilbert--Smith reduction says a locally compact group acting faithfully and continuously on a connected manifold, if not a Lie group, contains a topologically embedded copy of \(\Zp\) for some prime \(p\). Restricting the action to that subgroup keeps joint continuity and faithfulness, contradicting \(p\)-adic exclusion. Hence the group is a Lie group.
Sources
Dependencies
Let \(n \geq 1\) and \(p\) a prime. Every jointly continuous action of \(\Zp\) on a connected Hausdorff second-countable topological \(n\)-manifold without boundary has nonzero kernel. Consequently the action factors through a finite quotient of \(\Zp\).
Proof sketch. If the action were faithful, the invariant chart subset would carry a faithful orientation-preserving action of an open subgroup isomorphic to \(\Zp\), contradicting the chart contradiction. The kernel is closed as an intersection of point stabilizers, and every nonzero closed subgroup of \(\Zp\) is open, so the action factors through a finite quotient. No compactness, global orientability, or triangulability of the manifold is used.
Sources
Consequence
Let \(M\) be a manifold as in the Hilbert--Smith conjecture and \(f\) an almost-periodic homeomorphism of \(M\): the cyclic subgroup generated by \(f\) has compact closure \(K\) in \(\Homeo(M)\). Then some positive iterate of \(f\) is the time-one map of a continuous real flow: there are an integer \(m \geq 1\) and a continuous homomorphism \(\Phi : \R \to \Homeo(M)\) with \(\Phi(1) = f^m\).
Proof sketch. The compact abelian group \(K\) acts faithfully and jointly continuously on \(M\). Evaluation on a countable dense subset of \(M\) embeds \(K\) into a countable product of copies of \(M\), so \(K\) is second countable. By the Hilbert--Smith conjecture it is a compact abelian Lie group; its identity component is a torus of finite index, so \(f^m\) lies in it for some \(m \geq 1\), and the surjective exponential map of the torus places \(f^m\) on a one-parameter subgroup.