Diff(M) and connectedness 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.
 A: $M$ 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. 
A: Let $M$ be a compact oriented manifold.  The following hold if and only if $M$ is connected.
1) $\text{Diff}_0(M)$ is simple.  
This was proven by Thurston if $M$ is connected; see
MR1445290 (98h:22024) 
Banyaga, Augustin(1-PAS)
The structure of classical diffeomorphism groups. (English summary) 
Mathematics and its Applications, 400. Kluwer Academic Publishers Group, Dordrecht, 1997. xii+197 pp. ISBN: 0-7923-4475-8 
If $M$ is not connected, then $\text{Diff}_0(M)$ contains normal subgroups consisting of elements that fix some connected components and don't fix others.
2) $\text{Diff}_0(M)$ does not decompose as a direct product.
If $M$ is the disjoint union of submanifolds $M_1$ and $M_2$, then it is clear that $\text{Diff}_0(M) = \text{Diff}_0(M_1) \times \text{Diff}_0(M_2)$.
If $M$ is connected, then one can show that $\text{Diff}_0(M)$ does not decompose as a direct product by exhibiting elements $f \in \text{Diff}_0(M)$ whose centralizers consist only of $\langle 1, f, f^2, \ldots \rangle$.  There are many such constructions; for instance, see
MR0985855 (90i:58151a) 
Palis, J.(BR-IMPA); Yoccoz, J.-C.(F-POLY)
Rigidity of centralizers of diffeomorphisms. 
Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 1, 81–98. 
A famous conjecture of Smale says that such elements should in fact be generic.  This was recently proven by Bonatti-Crovisier-Wilkinson for $C^1$ diffeomorphisms; see
MR2511588 (2010g:37035) 
Bonatti, Christian(F-DJON-IM); Crovisier, Sylvain(F-PARIS13-AG); Wilkinson, Amie(1-NW)
The C1 generic diffeomorphism has trivial centralizer. (English summary) 
Publ. Math. Inst. Hautes Études Sci. No. 109 (2009), 185–244. 
