TITLE: Diff(M) and connectedness
QUESTION [9 upvotes]: Can anything be said about connectedness of a smooth manifold M from some property of Diff(M) in an analogous way Like C(X) has no idempotents iff X is connected.

REPLY [9 votes]: $M$ is connected if and only if the connected component of $Diff(M)$ (equivalently, of $Diff_c(M)$) acts transitively on $M$.
Edit: I just remembered, that the Lie algebra of compactly supported vector fields determines the base manifold up to diffeomorphism, see:
MR0064764 (16,331a) Reviewed 
Shanks, M. E.; Pursell, Lyle E.
The Lie algebra of a smooth manifold. 
Proc. Amer. Math. Soc. 5, (1954). 468–472. 
This is also true for larger Lie algebras, and for complex Stein manifolds, see:
MR0516602 (80g:57036) Reviewed 
Grabowski, J.
Isomorphisms and ideals of the Lie algebras of vector fields. 
Invent. Math. 50 (1978/79), no. 1, 13–33. 
Moreover, the group of compactly supported diffeomorphisms determines the base manifold completely, but I cannot find the relevant paper now.