I would like to ask questions about the definition of the Heegaard splitting. The following are the facts I know.
A Heegaard splitting says that any 3-manifold is built up from two handlebodies and a homeomorphism between boundaries of the handlebodies.
If $f$ and $g$ are isotopic such homeomorphisms, the 3-manifolds obtained are homeomorphic.
This is the fact what I know and want to prove it. But I don't know how to prove the second part.
How do I show that two isotopic homeomorphisms of boundaries of handlebodies produce the homeomorphic 3-manifolds?
Also, more generally, let $M$ and $M'$ be 3-manifolds with boundary. Suppose that $A\subset \partial M$ and $B \subset \partial M'$ are homeomorphic sub manifolds. Let $f:A \to B$ be a homeomorphism from $A$ to $B$. We glue $M$ and $M'$ via $f$. Does the homeomorphism class of the resulting manifold depend only on the isotopy class of the homeomorphism $f$?
Does the answer of the previous questions depend on what 3-manifolds I want to consider? Like, smooth, topological, piece-wise linear etc.
Edit: I am not familiar with ''collar'' in the comment below. I appreciate if one can explain more detail. I also want to know if collar exists for any type of manifolds.