Let $S_g$ be the closed, compact, orientable surface of genus g, which is defined up to homeomorphism. Let $MCG(S_g )$ be the mapping class group of $S_g$, which is defined as the isotopy classes of homeomorphisms of $S_g$. Let $H_g$ be the handlebody of genus g, which bounds $S_g$. A system of disks $D_1, \ldots, D_g$ are properly embedded disks in $H_g$ such that $\partial H_g - \cup_{i=1}^g \partial D_i $ is a 2-sphere with 2g deleted open disks. See for references, Jesse Johnson's lecture notes named "Notes on Heegaard Splittings". Consider the following question:
Let be $D_1,\ldots, D_g$ and $D'_1, \ldots , D_g'$ two systems of disks in $H_g$. Is there a homeomorphism of $\partial Hg = S_g$ which maps $\partial D_i$ to $\partial D_i'$ for all $i$.
Another related question is
Fix two systems of disks as above. Assume that two homeomorphisms $f$ and $g$ satisfies the property of the above question. Are $f$ and $g$ necessarily isotopic to each other?
Does something related to the above question hold?
Here, the context is Heegaard Splittings, which are obtained by glueing two handlebodies $H, H'$ of the same genus along their boundaries. The glueing map can be identified with an element of the mapping class group of the boundary surface. In some literature, there are two systems of disks given on $H'$ resp. $H$ . So I was wondering, if the glueing map is in any suitable way equivalent to two systems of disks.
Note that systems of disks are sometimes also called a complete system of meridian disks.
$\partial H_g\setminus \cup_{i=1}^g \partial D_i$ is a genus $0$ surface with $2g$ boundary circles and the same is true for $\partial H_g\setminus \cup_{i=1}^g \partial D_i^\prime$. From the classification of surfaces we know that they are homeomorphic. The homeomorphism $f$ must send boundary circles to boundary circles. We can assume that the homeomorphism sends the two copies of $\partial D_i$ to the two copies of $\partial D_i^\prime$. (This is because every permutation of boundary circles is realised by some self-homeomorphism of the 2-sphere.) Hence the homeomorphism $f$ yields a well-defined homeomorphism of $S_g$. (This is because $S_g$ is obtained from $S_g\setminus \cup_{i=1}^g \partial D_i$ by identifying the pairs of copies of $\partial D_i$.) This is the homeomorphism you want in your first question.
For the second question one has to note that self-homeomorphisms of the genus $0$ surface with $2g$ boundary circles are homotopic to each other as soon as one knows how they permute the boundary circles. So the answer should be yes.