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 $M=T^{n-2}\times S^2$ we have that the group $\pi\_i(Diff(M))$ has infinite rank for any $n\geq 5$ and $0\leq i\leq n-3$. 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? 



Manifolds whose diffeomorphism group has the homotopy type of a manifold itself
If $M$ is a surface of genus $g\ge 2$, then the Earle-Eells Theorem asserts that the connected components of Diff(M) are contractible, so Diff(M) has the homotopy type of a non-compact 0-dimensional manifold.


The same is true for hyperbolic manifolds of dimension 3, except that there are only finitely many components then. ([Gabai](https://projecteuclid.org/euclid.jdg/1090348284))


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.


