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Smoothing finite group actions on three-manifolds. (English) Zbl 1469.57027

Whereas by classical results, every continuous action (i.e., by homeomorphisms) of a finite group on a manifold of dimension at most 2 is conjugate to a smooth action, there are many examples of finite group actions on 3-manifolds which are not conjugate to smooth actions (actions with wild fixed points sets). The strong main result of the present paper answers an old and natural question, stating that every continuous action of a finite group \(G\) on a smooth, compact or non-compact 3-manifold \(M\) is a uniform limit (i.e., in the uniform topology) of smooth actions of \(G\) on \(M\) (in particular, no “exotic finite groups” \(G\) occur which admit a continuous but no smooth action on a 3-manifold). As the author notes, in higher dimensions there exist even free finite group actions which are not uniformly approximable by smooth actions.
The proof of the main result starts with the case of free actions which is a consequence of classical results of Bing and Moise, then proceeds for general actions in three steps by smoothing actions on successively larger subsets of \(M\) (considering fixed point sets and their complements; for sketches of the proofs of these steps, see the introduction of the paper).

MSC:

57M60 Group actions on manifolds and cell complexes in low dimensions
55M35 Finite groups of transformations in algebraic topology (including Smith theory)
57M30 Wild embeddings
57R10 Smoothing in differential topology
57S10 Compact groups of homeomorphisms
57S17 Finite transformation groups
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