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> What type of compact manifolds, can be acted freely by symmetric group $S\_{m}$ for some $m>2$?
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> Is there a compact manifold which can be act freely by all symmetric groups $S\_{m}$?
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This question have been asked already [here](https://math.stackexchange.com/questions/981053/free-action-of-symmetric-group)
and is indirectly related to [this](https://mathoverflow.net/questions/184814/a-possible-generalization-of-the-borsuk-ulam-theorem-via-action-of-symmetric-gro).



Free action of symmetric groups
The answer to the second question is more or less no by Theorem 2 in Popov's [Finite subgroups of diffeomorphism groups](http://arxiv.org/abs/1310.6548): for every compact connected smooth manifold $M$ there is a constant $b\_M$ such that if the alternating group $A\_n$ acts (by diffeomorphisms, but not necessarily freely) on $M$, then $n \le b\_M$. 


I don't really understand what kind of answer you're expecting to the first question. 


