Suppose $S^3$ is PL sphere on which a finite group $G$ acts by PL homeomorphisms. Is it always possible to find a compatible smooth structure such that $G$ acts by diffeomorphisms?

I am not quite sure how smoothing works for trivial $G$, but maybe this can be generalized to the situation described above?


I think the answer to your question is "always". You may want to look at the following paper:

Kwasik, Sławomir; Lee, Kyung Bai. Locally linear actions on $3$-manifolds. Math. Proc. Cambridge Philos. Soc. 104 (1988), no. 2, 253--260. MR0948910.

In particular, Corollary 2.2: A topological action of a finite group G on a closed 3-manifold M is smoothable if and only if it is simplicial (in some triangulation of M).


The answer is "always", but it is not proven in the paper by Kwasik and Lee. They only show the existence of an equivariant smooth structure which is not necessarily compatible with the PL structure. The statement holds for manifolds up to dimension four. I tried to write a readable proof and put it on the arXiv.

  • 1
    $\begingroup$ Christian Lange, I have a follow-up question about your very helpful comment. If in addition to the PL action of G on M we are given a submanifold S of M which is G-invariant, can we choose the compatible G-equivariant smooth structure on M in such a way that S becomes a smooth submanifold? This is like asking whether PL orbifold pairs can be smoothed. $\endgroup$ – Peter Shalen Aug 9 '17 at 22:32

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