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.



Diff(M) and connectedness
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. 


