This is false for $n=1.$ The mapping class group of the torus is $SL(2, Z),$ of which the homeomorphisms you describe are but a small part - the parabolic matrices $\begin{pmatrix}1 & n\\ 0 &1\end{pmatrix},$ unless I am very confused. I cautiously believe the statement is true for $n=2,$ by <cite authors="Allen Hatcher" mrnumber="420620" cite="_Topology_ **15** (1976), no. 4, 343--347">_Allen Hatcher_, MR 420620 [**Homeomorphisms of sufficiently large $P^{2}$-irreducible $3$-manifolds**](http://www.ams.org/mathscinet-getitem?mr=420620), _Topology_ **15** (1976), no. 4, 343--347.</cite> *UPDATE* for $n=3,$ the best I can find is: <cite authors="Richard Stong and Zhenghan Wang" mrnumber="1769331" cite="_Topology Appl._ **106** (2000), no. 1, 49--56">_Richard Stong and Zhenghan Wang_, MR 1769331 [**Self-homeomorphisms of 4-manifolds with fundamental group Z**](http://dx.doi.org/10.1016/S0166-8641(99)00076-0), _Topology Appl._ **106** (2000), no. 1, 49--56.</cite> Which classifies up to pseudo-isotopy (NOT isotopy), and does give the expected result, as far as I can tell.