[Homotopy groups of Lie groups](https://mathoverflow.net/questions/8957/homotopy-groups-of-lie-groups/8996#8996)
I asked it also there, and I still don't know the answer, so I try again.


I would like to know a closed manifold (possibly of low dimension) such that $\pi\_2(\textrm{Diff}(M))\neq 0$.



nontrivial $\pi_2(\textrm{Diff}(M))$
$\newcommand{\Diff}{\mathrm{Diff}}$Probably the simplest such manifold is $S^1 \times S^2$, whose diffeomorphism group has the homotopy type of $O(2) \times O(3) \times \Omega SO(3)$. This has $\pi\_2$ equal to $\pi\_2\Omega SO(3)=\pi\_3 SO(3) = {\mathbb Z}$. The $\Omega SO(3)$ term is realized by rotating the $S^2$ slices of $S^1\times S^2$ by an element of $SO(3)$ that varies as one goes around the $S^1$ factor of $S^1 \times S^2$. This calculation was originally done in a paper of mine:


* *On the diffeomorphism group of $S^1\times S^2$*, Proc. Amer. Math. Soc. 83 (1981), 427-430, doi:[10.1090/S0002-9939-1981-0624946-2](https://doi.org/10.1090/S0002-9939-1981-0624946-2).


An updated version of this paper is [posted on my webpage](https://pi.math.cornell.edu/%7Ehatcher/Papers/newDiffS1xS2.pdf).


**Added later:**


In higher dimensions, spheres provide interesting examples. By an elementary argument there is a homotopy equivalence $\Diff(S^n) \simeq O(n+1) \times \Diff\_\partial(D^n)$ for all $n$, where the subscript $\partial$ denotes diffeomorphisms that restrict to the identity on the boundary of $D^n$. The question is whether $\Diff\_\partial(D^n)$ is contractible. The status of this is: true for $n\le3$, unknown for $n=4$, and false for each $n\ge 5$. The noncontractibility can be deduced from the sequence $$\cdots\ \to \pi\_2 \Diff\_\partial(D^{n-2})\to \pi\_1 \Diff\_\partial(D^{n-1})\to \pi\_0 \Diff\_\partial(D^n) = \Theta\_{n+1}$$ where $\Theta\_{n+1}$ is the group of exotic (n+1)-spheres and the equality $\pi\_0 \Diff\_\partial(D^n) = \Theta\_{n+1}$ is assuming $n\ge 5$. Usually $\pi\_0 \Diff\_\partial(D^n)$ is nonzero since exotic spheres exist in most dimensions greater than $6$, and in the rare dimensions in which they don't exist one can appeal to known results about how far some elements of $\Theta\_{n+1}$ pull back in the sequence above. For example, Cerf's pseudoisotopy theorem says the map from $\pi\_1 \Diff\_\partial(D^{n-1})$ to $ \pi\_0 \Diff\_\partial(D^n)$ is surjective for all $n\ge 6$, so in particular $\pi\_1 \Diff\_\partial(D^5)$ is nonzero since it maps onto $\Theta\_7={\mathbb Z}/28$. A paper of Crowley and Schick posted on the arXiv last month shows there is a nontrivial element of $\Theta\_{8k+2}$ that pulls all the way back to $\pi\_{8k-6} \Diff\_\partial (D^7)$, hence $\Diff\_\partial(D^n)$ has infinitely many nontrivial homotopy groups for all $n\ge 7$.


Since the groups $\Theta\_{n+1}$ are finite (apart perhaps from the unknown $\Theta\_4$), these constructions don't give nontrivial rational homotopy groups of $\Diff\_\partial(D^n)$, but there is another construction that does, coming from algebraic K-theory. Using Waldhausen's big machine, [Farrell and Hsiang in 1978](https://mathscinet.ams.org/mathscinet-getitem?mr=520509) computed $\pi\_i \Diff\_\partial (D^n) \otimes{\mathbb Q}$ in a stable range $n\gg i$ to be ${\mathbb Q}$ for $i\equiv 3$ mod $4$ and $n$ odd, and $0$ otherwise. (This is the result mentioned in Vitali Kapovitch's answer.)


