In a paper by John Franks I stumbled upon the following:
Let $M$ be a surface and $f:M \rightarrow M$ be a homeomorphism, which is homotopic to the identity on $M$. That means, that there is continuous map $h: [0,1] \times M \rightarrow M$, such that $h(0,\cdot) = id_M$ and $h(1,\cdot)= f$.
If $M$ has negative Euler characteristic, then the homotopy is unique up to homotopy. (i.e. if there is another homotopy $g: [0,1] \times M \rightarrow M$, such that $g(0,\cdot) = id_M$ and $g(1,\cdot)= f$, then $h$ and $g$ are homotopic).
In Frank's paper, the surface $M$ is additionally equipped with a riemannian metric of constant negative curvature and of genus zero. I don't know if this is essential.
Currently, I have no idea why this is true. Can anyone explain or point me to a reference, where this is proved?
Thanks alot for the hints! I found the following proposition in the Algebraic Topology Hatcher:
Prop 1B.9 Let $X$ be a connected CW complex and let $Y$ be a $K(G,1)$ space. Then every homomorphism $\Pi_1(X,x_0) \rightarrow \Pi_1(Y,y_0)$ is induced by a map $(X,x_0) \rightarrow (Y,y_0)$ that is unique up to homotopy fixing $x_0$.
Now, our maps $g,h$ are both homotopic to the identity and hence induce the same homomorphism on the fundamental group. It follows from the proposition that $g$ and $h$ are homotopic.