Infinite collection of homeophisms from $S^1\times S^1$ to $S^1\times S^1$ that are not homotopic. While studying for my topology exam I tried to come up with a solution of the following problem.
Let $X=S^1\times S^1$. Give (with proof) an infinite collection of homeomorphisms $f_i:X\to X$ such for all $i\neq j$ we have that $f_i$ and $f_j$ are not homotopic.
I was thinking about the identity map and the map that cuts the torus and twists the right part and put it together. Like

Can someone tell me if this is a good approach, if not give an example?
The next thing I really have a problem with is how to proof that these maps are not homotopic.
Thanks!

First note that it is easy to distinguish selfmaps of the torus by means of the fundamental group.  Thus, rather than considering selfmaps of $X$, consider maps $S^1\to X$.  These are classified up to homotopy by $\pi_1(X)=\mathbb{Z}^2$, an infinite group.  Now to obtain infinitely many selfmaps of $X$, consider the projection to the first factor $X\to S^1$ and compose it with the maps $S^1\to X$ as above.
If you need specifically homeomorphisms, consider the action of $SL(2,\mathbb{Z})$ on the plane.  This defines a self-map of the torus by passing to the quotient $\mathbb{R}^2/\mathbb{Z}^2$.  By examining the action on the first factor, you can similarly show that there are infinitely many of these that are not homotopic.
