# Heegaard splitting and mapping class group

I would like to ask questions about the definition of the Heegaard splitting. The following are the facts I know.

1. A Heegaard splitting says that any 3-manifold is built up from two handlebodies and a homeomorphism between boundaries of the handlebodies.

2. 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.

1. How do I show that two isotopic homeomorphisms of boundaries of handlebodies produce the homeomorphic 3-manifolds?

2. 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 $f$?

3. 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.

• I believe what you do is choose a collar of the boundary and perform the isotopy along the collar. – Tim kinsella Nov 5 '14 at 20:33
• In the smooth category this is a quite general result: gluing manifolds along isotopic diffeomorphisms gives diffeomorphic manifolds. Have a look at Hirsch, Differential Topology, chapter 8 sections 1 and 2, especially theorem 2.3 – Lor Nov 5 '14 at 20:59
• Same in the topological (and PL) category; works in all dimensions too. – Moishe Kohan Nov 7 '14 at 19:51
• For the concept of a collar, see Lee's Introduction to Smooth Manifolds. – Tim kinsella Nov 7 '14 at 23:33

The comments above and the references they contains are good answers in my opinion.

Let me just add a picture of how to build an explicit homeo, given two isotopic gluings. As Tim kinsella says, use collars and the cylinder given by the isotopy.

I hope this very bad picture can help.

The first row of the picture tells you how to build a manifold inserting between $M_1$ and $M_2$ a cylinder $\partial M_1\times [0,1]$. This is the cylinder of the isotopy of the two gluings $f_1$ and $f_2$. Say that you glue on the left via $f_1$ and on the right via $f_2$. Call this manifold $\hat M$

The two bottom part of the picture says that you can "insert" the cylinder of the isotopy in a collar of $\partial M_1$ as well as in a collar of $\partial M_2$.

So the second line is the result of gluing $M_1$ and $M_2$ via $f_2$ and the third is the result of gluing $M_1$ and $M_2$ via $f_1$.

Both are homeo to $\hat M$.

• Thank you for a good explanation. Could you answer the question part 2 as well? I am not sure if I can use the method above for restricted collar to submanifold in the boundary. – Snow Nov 12 '14 at 15:53
• as far as you have collars (you need collars of $\partial A$ in $\partial M$), then yes the same construction works. (b.t.w. what do you mean exactly by submanifold of $\partial M$? is just an open subset of $\partial M$? or a compact sub-manifold with boundary of $\partial M$, or a connected component of $\partial M$? also, I guess $B\subset \partial M'$, right?) In low dimension (1,2, and 3) there are no problems of existence of collars (in the case $A$ is a compact submanifold of $\partial M$) and the three categories: Top, Pl, Smooth coincide. Problems can arise in higer dimension. – user126154 Nov 12 '14 at 16:32
• Yes $B \subset \partial M$. I fixed it. By submanifold of $\partial M$, I meant an embedding A into $\partial M$. Does it make sense? – Snow Nov 12 '14 at 20:02
• What I confuse in Question 2 is that if if $A \subset \partial M$ is not all of $\partial M$, it looks like for me the isotopy cannot be absorbed in $M$ nicely? – Snow Nov 12 '14 at 20:04