I am trying to have a better understanding of how one goes , "travels" between the different formats/layouts of open books for a fixed given 3-manifold M; between the abstract type and the "actual" type. I am new to this, and I have not been able to find clear explanations, or whether there are canonical choices that help make things more clear. Please critique/suggest :

The abstract type is given by ( $\Sigma, f$) , where $\Sigma$ is a compact, oriented surface with non-empty boundary, and $f$ is an element of $MCG( \Sigma$), the Mapping Class Group of $\Sigma$ , and the "actual" type ($ B, \pi$ ), where $B$ is a fibered link, and $\pi$ is a map , so that $\pi: (M-B) \rightarrow S^1$ is a locally-trivial bundle with fiber the surface $\Sigma$, so that $ \partial \Sigma =B$ , and $\pi|_N(B)$, (where N is a tubular neighborhood of B --a fibered link has a trivial normal bundle) is the argument map . It seems reasonable to assume here that $\Sigma$ is a Seifert surface for the fibered link $B$.

So, let me lay out what I think about the two transitions:

1) Between $(B, \pi)$ and $(\Sigma_\pi, f_\pi$ ).

The ambiguity I see here is that , assuming $\Sigma$ is a Seifert surface for $B$, does not narrow $\Sigma$ down even up to homeomorphism, let alone isotopy, because there are many possible Seifert surfaces for any $B$ unless maybe we fix the genus of $\Sigma$ ( So that $\Sigma$ is given up to boundary components by $B$ and up to genus , so we have $\Sigma_{n,g}$ up to homeomorphism ). Is there a canonical way of choosing $\Sigma$ here?. We could , I guess just define $ \Sigma_\pi:= \pi^{-1} (\theta)$ , for any $\theta \in S^1 $ . But, how does one determine $f_\pi$? I know we could calculate the total monodromy of $\pi$ , by choosing a Riemann metric g, computing the dual vector field to $d\pi$, and then flowing it. Is this how this is usually done?

Now, for the opposite direction,

2)Between $( \Sigma, f )$ and $( B, \pi )$

The mapping torus $M_f$ clearly gives us the map $\pi$ onto $S^1$, by following along the copy of $S^1$ in the mapping torus of $f$ along $\partial (S^1 \times D^2)$. And, if $\Sigma$ must be a Seifert surface for B, then we would just let $B:= \partial \Sigma$. Is this how things are usually done?

An additional question, please:

3) Why do we define equivalence AOBs $( \Sigma_1, f_1)$ and $(\Sigma_2, f_2)$ to be equivalent if there exists a homeo. $h: \Sigma_1 \rightarrow \Sigma_2$ satisfying :

$h \circ f_1 = f_2 \circ h$ ?

This last shows that $f_1$ and $f_2$ are conjugate elements in $MCG(\Sigma)$. Does this last condition guarantee that $f_1, f_2$ are isotopic $(in MCG(f)$? ; since $f_1, f_2$ are isomorphic, set $f =f_1= f_2)$?