I have a very stupid question. Let $M$ be a closed smooth manifold. In particular cases the homotopy type of the diffeomorphism group $Diff(M)$ can be very pathological. For example, in the case of $M=T^{n2}\times S^2$ we have that the group $\pi_i(Diff(M))$ has infinite rank for any $n\geq 5$ and $0\leq i\leq n3$. But my question is about an opposite situation. Does someone know when the diffeomorphism group $Diff(M)$ has the homotopy type of another closed smooth manifold? Or a more simple question, when $Diff(M)$ is at least a Poincare space?

1$\begingroup$ There appear to be very few such cases, and they are only in low dimensions. ThiKu's response covers many of them, but of course most geometric 3manifolds satisfy your condition. For higherdimensional manifolds the answer could very well be never, as far as I know. $\endgroup$ – Ryan Budney Jul 4 '16 at 18:27

$\begingroup$ @RyanBudney Yes, I am more interesting in higher dimensional manifolds. And, please, could you say what kind of reason for failing $Diff(M)$ to be a manifold in this case? Is there a computation of the cohomology ring $H^*(Diff(M),\mathbb{Q})$? Or is there a more elementary argument? $\endgroup$ – Nikolay Konovalov Jul 4 '16 at 18:58

$\begingroup$ Yes, there are some results of that kind. They go back to the work of Antonelli, Burghelea and Kahn. I believe they show that cohomology is infinitelygenerated, but it's been a while since I've looked at their paper. P.L. Antonelli, D. Burghelea, P.J. Kahn, "The nonfinite homotopy type of some diffeomorphism groups" Topology , 11 : 1 (1972) pp. 1–49 $\endgroup$ – Ryan Budney Jul 4 '16 at 19:10
If $M$ is a surface of genus $g\ge 2$, then the EarleEells Theorem asserts that the connected components of Diff(M) are contractible, so Diff(M) has the homotopy type of a noncompact 0dimensional manifold.
The same is true for hyperbolic manifolds of dimension 3, except that there are only finitely many components then. (Gabai)
Moreover, it is a Theorem of Smale that $Diff(S^2)$ deformation retracts to $SO(3)$. The analogous statement that $SO(4)$ is a deformation retract of $Diff(S^3)$ was known as the Smale conjecture and was proved by Hatcher.

$\begingroup$ O, thanks. It's nice examples. And of course the same statement is true in the case of the torus $T^2$. The connected component $Diff_0(T^2)$ is homotopy equivalent to $T^2$ itself. $\endgroup$ – Nikolay Konovalov Jul 4 '16 at 18:51

$\begingroup$ As I understand it, Milnor found that $Diff(S^6)$ has 28 components, so in particular isn't homotopic to $SO(7)$. $\endgroup$ – Allen Knutson Jul 24 '16 at 21:43