The question asks whether a smooth manifold ( M ) that admits no effective smooth actions by non-trivial finite groups must also admit no effective continuous actions by non-trivial finite groups.

To address this, we need to consider the distinction between smooth and continuous group actions. A smooth action requires each group element to act as a smooth diffeomorphism, while a continuous act...

No in general (in dimensions ≥ 3).

There are plenty of continuous (topological) finite group actions on smooth manifolds that are not topologically conjugate to smooth actions. The classical source of such examples are wild actions, whose fixed point sets are wildly embedded and hence cannot occur for smooth actions (where fixed point sets are smoothly embedded submanifolds).

Concrete mechanism:...

This is an excellent question about the relationship between smooth and continuous group actions on manifolds. The answer is no - there exist smooth manifolds that admit continuous but not smooth effective actions by finite groups.

Counterexamples

The key insight is that there are topological actions that cannot be smoothed. Here are the main examples:

1. Bing-Borsuk Examples

Some of...

No.

For dimensions ≥ 3 one can find smooth manifolds that

(a) do not admit any non-trivial smooth action of a finite group, but
(b) do carry an effective continuous action of a finite group.

Below is an explicit construction. (After the construction we explain why the phenomenon cannot occur in dimension 1 or 2.)

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1. A man...

<think> The question is about smooth manifolds that do not admit any effective smooth actions of finite groups except the trivial group. The question asks whether such a manifold also admits no continuous effective actions of finite groups. In other words, if there are no smooth finite group actions, does that imply there are no continuous ones either?

First, recall that an effective action means...

On Manifolds and Finite Group Actions

This is an interesting question at the intersection of differential topology and transformation groups. Let me analyze whether a smooth manifold that doesn't admit smooth actions of finite groups necessarily doesn't admit continuous actions either.

Key Considerations

The relationship between smooth and continuous group actions is subtle. While every sm...

No – beginning in dimension 3 the two conditions are independent.
There are smooth manifolds which have

 • no non–trivial smooth (i.e. ) actions of finite groups, but
 • do admit non–trivial continuous actions of finite groups.

Consequently the absence of smooth actions does ​not​ imply the absence of continuous actions.

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Here is a brief summary of what is known and why, in general, “no smooth finite–G–action” does not force “no continuous finite–G–action.” In fact one really needs to rule out topological actions, because there are plenty of “wild’’ finite‐group actions on smooth manifolds which are not conjugate to any smooth action.

1. The naive “metric‐averaging’’ argument fails to produce a smooth i...

Excellent question. This gets to the heart of a classic and powerful result in the theory of transformation groups.

The answer to your question is yes. For a smooth manifold , the following two statements are equivalent:

1. admits no effective smooth action by any non-trivial finite group.
2. admits no effective continuous action by any non-trivial finite group.

Clearly...