---
tag: DS2AS9TSZP
kind: conjecture
name: Hilbert–Smith conjecture
msc: ["57S05", "57S10", "22A05"]
uses: [AHV05ZBCYW, 7JBRGCVJRB, QQQ9B1DK4P, 0P7G9NGG2G]
about: [AHV05ZBCYW, QQQ9B1DK4P]
sources:
  - url: https://en.wikipedia.org/wiki/Hilbert%E2%80%93Smith_conjecture
same_as:
  wikidata: Q5761258
  wikipedia: Hilbert–Smith conjecture
history:
  posed_by: "Paul A. Smith"
  first_stated: "1941"
  notes: "A strengthening of the fifth problem of David Hilbert (1900)."
---

## Statement

Let $M$ be a connected [[topological manifold|QQQ9B1DK4P]] and let $G$ be a [[locally compact group|AHV05ZBCYW]].
Suppose that $G$ acts on $M$ continuously and [[faithfully|0P7G9NGG2G]].
That is, the map $G \times M \to M$ is continuous, and only the identity of $G$ fixes every point of $M$.
Then $G$ is a [[Lie group|7JBRGCVJRB]].

## Remarks

The conjecture is a form of [[Hilbert's fifth problem|NR70GHZ37D]], which Hilbert stated for transformation groups; it is named after David Hilbert and Paul A. Smith.
It is known to be equivalent to the claim that, for every prime $p$, the group $\Z_p$ of $p$-adic integers cannot act continuously and faithfully on a connected manifold.
The conjecture is proved for manifolds of dimension at most $3$; the case of dimension $3$ is due to John Pardon.
It is also proved for actions by Lipschitz maps and for actions by diffeomorphisms.
In general it is open.

## Explanations

### L1 human

A group can act on a space by moving its points around, as rotations move the points of a sphere.
The conjecture says that if a group acts on a manifold in a continuous and faithful way, the group must be a smooth Lie group.
In other words, exotic groups such as the $p$-adic integers should not act continuously on a manifold unless some element other than the identity leaves every point in place.