Fix $n$ odd and let $M=S^n\times S^n$. The diffeomorphism group Diff($M$) acts on the homology group $H_n(M)\simeq \mathbb Z^2$ inducing a surjection $d: \text{Diff}(M) \rightarrow \text{SL}(2,\mathbb Z)$. I’d like a recipe for constructing an explicit diffeomorphism $f$ so that $d(f)=A$, where $A= \left(\begin{array}{ll}1&1\\0&1\end{array}\right)$.
In the case $n\ge 3$, the fact that $d$ is surjective can be found in
- Wall, Classification of $(n-1)$-connected $2n$-manifolds, Corollary to Theorem 1, or
- Wall, Classification problems in differential topology II, Lemma 10.
However, the argument (which applies when $M$ is any $(n-1)$-connected $2n$ manifolds for $n\ge3$) goes through something like the h-cobordism theorem, so it's not explicit. I'm hoping there is a more concrete description in the special case $M=S^n\times S^n$.
In the case $n=1$, one can use that SL$(2,\mathbb Z)$ acts on the plane $\mathbb R^2$ preserving the integer lattice $\mathbb Z^2$, inducing an action on $\mathbb R^2/\mathbb Z^2\simeq S^1\times S^1$. Alternatively, the diffeomorphism $A\in\text{Diff}(S^1\times S^1)$ is isotopic to a Dehn twist $T$ about the meridian curve (or maybe it’s the longitude), and $T$ is supported on an annulus, where it can be described by the formula \begin{array}{ccc} T:&S^1\times[1,2]&\rightarrow& S^1\times[1,2]\\ &(\theta, r)&\mapsto&(\theta+2\pi r, r).\end{array}
It's not clear to me how/if either of these descriptions generalize to higher dimensions. Any reference or suggestion would be appreciated!