When gluing maps are isotopic?

Question

Let $M$ and $M'$ be compact orientable connected topological 3-manifolds. (One may need more conditions to answer the question.)

Suppose we have two homeomorphisms $f$ and $g$ from the boundary surface $\partial M$ to the boundary surface $\partial M'$. Consider the gluings of $M$ and $M'$ along $f$ and $g$ respectively.

Assume that the resulting manifolds $M\cup_f M'$ and $M\cup_g M'$ are homeomorphic.

Question: What can we say about a relation of $f$ and $g$? Are they isotopic?

Answer 1

To give one possible elaboration of studiosus's counterexample, let $M = M' = S^1 \times D^2$, let $f,g\colon \partial M \to \partial M'$ be given by $(x,y) \mapsto (x,y)$ and $(-x,y) \mapsto (x,-y)$. These maps are not isotopic (think what they do to the fundamental group of $\partial M$), but both $M \cup_f M'$ and $M \cup_g M'$ are $S^1 \times S^2$.