Let $M$ be a topological/differentiable manifold. Is there any topology on the group of homeomorphisms/diffeomorphisms with compact support, turning it into a (locally-)compact topological group?

(My question is motivated by the fact that the isometries of a (locally-)compact metric space with finitely many connected components form a (locally-)compact topological group, and I'd like to see how far this analogy can be pushed.)


You can always make any group locally compact by giving it the discrete topology, but I doubt that's what you want. Then the diffeo version of your question is up against the Montgomery-Zippin theorem that $\mathrm{Diff}_c(M)$, endowed with any locally compact topology, would contain no small subgroups and therefore be a finite-dimensional Lie group. Which it isn't (unless made discrete). See:

Bochner & Montgomery, Locally compact groups of differentiable transformations (1946),

Montgomery & Zippin, Topological transformation groups (1955), Thm 2, p. 208,

Montgomery, Finite dimensionality of certain transformation groups (1957).


The group $\mathrm{Diff}_c(M)$ is in a natural way a Lie group modeled on nuclear (LF) spaces (like the space of test functions). This was shown in

  • Peter W. Michor: Manifolds of smooth maps II: The Lie group of diffeomorphisms of a non compact smooth manifold. Cahiers Topologie Geometrie Differentielle 21 (1980), 63--86.

The connected component of this group is simple (Thurston).

To the answer of Francois Ziegler one can add the following theorem of Omori: If a Banach Lie group $G$ acts faithfully on a compact manifold (or a noncompact finite dimensional one via diffeomorphisms with compact support), then $G$ is finite dimensional.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.