TITLE: Self homeomorphisms of $S^2\times S^2$
QUESTION [12 upvotes]: Every matrix $A\in SL_2(\mathbb{Z})$ induces a self homeomorphism of $S^1\times S^1=\mathbb{R}^2/\mathbb{Z}^2$. For different matrices these homeomorphisms are not homotopic, as the induced map on $\pi_1(S_1\times S_1)$ is given by $A$ (w.r.t the induced basis).
So I am wondering, whether a similar construction also works for other spheres than $S^1$. So is there any non-obvious self-homeomorphism of $S^2\times S^2$ (I know i can use degree $-1$ maps in any choice of coordinates / flip the coordinates /compose such maps).

REPLY [3 votes]: A small comment extending Greg's comments, given an element of $[g] \in \pi_n Diff(S^n)$ you can construct a diffeomorphism 
$f : S^n \times S^n \to S^n \times S^n$ by sending the pair $(x,y)$ to $(x,g_x(y))$, so the corresponding matrices would be upper-triangular as in Greg's examples. 
$Diff(S^n)$ has the homotopy-type of $O_{n+1} \times Diff(D^n)$. 
Greg's examples come from situations where there are non-trivial elements of $\pi_n O_{n+1}$ that you "know" because they're coming from splittings of the fibrations $SO_n \to SO_{n+1} \to S^n$, but you could also get non-trivial elements from finding elements in $\pi_n Diff(D^n)$. My understanding is there's little to nothing known about these homotopy groups. The rational homotopy $\pi_j Diff(D^n)\otimes \mathbb Q$ of $Diff(D^n)$ is only understood in the "Igusa Stable Range" of $j \leq \min\{ \frac{n-4}{3}, \frac{n-7}{2} \}$ so it's not useful for the kind of constructions you'd like.