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Let $M$ be a smooth finite dimensional manifold, how restrictive is it to require $M$ to admit a smooth action by a finite dimensional Lie group $G$?

Related questions/approaches: Of course we need $\mathrm{dim}(G) \geq \mathrm{dim}(M)$, are there any results relating the minimal dimension of a Lie group acting transitively to that of $M$, perhaps in special cases?

Going in the other direction, any criteria which easily allow to say that a given smooth manifold does not admit a transitive group action?

EDIT: in the first question I meant to write transitive smooth group action.

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    $\begingroup$ In the first question, you have to specify what sort of action you want. Besides the trivial action, every manifold admits vector fields and hence non-trivial $\mathbb{R}$ actions. $\endgroup$
    – alvarezpaiva
    May 12, 2021 at 15:55
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    $\begingroup$ The answer is "extremely special". There are plenty of restrictions on these manifolds. Take a look at the $2$ and $3$ dimensional cases, for example. But there is a long history to this question, going back to the 60's. $\endgroup$
    – Ryan Budney
    May 13, 2021 at 2:52

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I suppose you want the action to be transitive as your title suggests. In this case, a classical theorem of Mostow (in 1950 for surfaces, in 2005 in general) says that for a compact homogeneous space $M = G/H$ the Euler characteristics is non-negative.

Mostow, G.D.: The extensibility of local Lie groups of transformations and groups on surfaces. Ann. Math. (2) 52, 606–636 (1950)

Mostow, G.D.: A structure theorem for homogeneous spaces. Geom. Dedic. 114, 87–102 (2005)

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