Let $X,Y$ be two compact, smooth, orientable 3 manifolds, each with an incompressible boundary component diffeomorphic to some genus $g $ surface $S_g$. Under an orientation-reversig diffeomorphism $f:S_g \to S_g$, those two manifolds can be glued together to obtain a new smooth, orientable manifold $X \cup_f Y$. I wonder now in how far the diffeomorphism type of this result depends on the choice of $f $. Using a collar, one can show that if $f $ and $g $ are two isotopic diffeomorphisms of $S_g $, then the corresponding gluings are diffeomorphic . Is this also a necessary condition ?
Some thoughts I have made so far: 1) If both X and Y are irreducible, then so is $X \cup_f Y$, and if $X \cup_f Y $ still has some boundary component, it must be an aspherical manifold. Hence, all important Information is contained in the fundamental group. Is it true that the homeomorphism type of aspherical 3 - manifolds obtained this way is already determined by their fundamental group ?
2) Also, I wonder if its true that the isotopy type of two orientation-reversing diffeomorphisms on a surface $S_g$ is determined by their action on the fundamental group. Edit: This is probably false, since mapping class groups of surfaces are very distinct from the corresponding fundamental group.
Any help is appreciated.
Edit: I apologize for missing and/or inaccurately placed capital letters. I wrote this question yesterday evening on my phone, and it was impossible to fix all the mistakes made by auto-correct (I am writing on a german phone).
Edit 2: I have also updated this question, according to what has already been solved by the answers and what is still open.