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.

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(1PAS) The structure of classical diffeomorphism groups. (English summary) Mathematics and its Applications, 400. Kluwer Academic Publishers Group, Dordrecht, 1997. xii+197 pp. ISBN: 0792344758 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.(BRIMPA); Yoccoz, J.C.(FPOLY) 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 BonattiCrovisierWilkinson for $C^1$ diffeomorphisms; see MR2511588 (2010g:37035) Bonatti, Christian(FDJONIM); Crovisier, Sylvain(FPARIS13AG); Wilkinson, Amie(1NW) The C1 generic diffeomorphism has trivial centralizer. (English summary) Publ. Math. Inst. Hautes Études Sci. No. 109 (2009), 185–244. 


$M$ 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. 

