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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.)

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2 Answers 2

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).

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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.

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